diff --git a/cirq-core/cirq/contrib/paulistring/optimize_test.py b/cirq-core/cirq/contrib/paulistring/optimize_test.py
index c5b14faf68c..235683852f9 100644
--- a/cirq-core/cirq/contrib/paulistring/optimize_test.py
+++ b/cirq-core/cirq/contrib/paulistring/optimize_test.py
@@ -50,14 +50,15 @@ def test_optimize():
 
     cirq.testing.assert_allclose_up_to_global_phase(c_orig.unitary(), c_opt.unitary(), atol=1e-6)
 
+    # TODO(#5546) Fix '[Z]^1' (should be 'Z')
     cirq.testing.assert_has_diagram(
         c_opt,
         """
-0: ───X^0.5────────────@────────────────────────────────────────
+0: ───X^0.5────────────@──────────────────────────────────────────────
                        │
-1: ───@───────X^-0.5───@───@────────────────@───Z^-0.5──────────
+1: ───@───────X^-0.5───@───@────────────────@───Z^-0.5────────────────
       │                    │                │
-2: ───@────────────────────@───[X]^(-7/8)───@───[X]^-0.25───Z───
+2: ───@────────────────────@───[X]^(-7/8)───@───[X]^-0.25───[Z]^(1)───
 """,
     )
 
diff --git a/cirq-core/cirq/protocols/json_test_data/CZTargetGateset.json b/cirq-core/cirq/protocols/json_test_data/CZTargetGateset.json
index be77ff8faea..23f3615b41d 100644
--- a/cirq-core/cirq/protocols/json_test_data/CZTargetGateset.json
+++ b/cirq-core/cirq/protocols/json_test_data/CZTargetGateset.json
@@ -8,5 +8,43 @@
     "cirq_type": "CZTargetGateset",
     "atol": 1e-08,
     "allow_partial_czs": true
+  },
+  {
+    "cirq_type": "CZTargetGateset",
+    "atol": 1e-06,
+    "allow_partial_czs": true,
+    "additional_gates": [
+      {
+        "cirq_type": "GateFamily",
+        "gate": {
+          "cirq_type": "ISwapPowGate",
+          "exponent": 0.5,
+          "global_shift": 0.0
+        },
+        "name": "Instance GateFamily: ISWAP**0.5",
+        "description": "Accepts `cirq.Gate` instances `g` s.t. `g == ISWAP**0.5`",
+        "ignore_global_phase": true,
+        "tags_to_accept": [],
+        "tags_to_ignore": []
+      },
+      {
+        "cirq_type": "GateFamily",
+        "gate": "XPowGate",
+        "name": "Type GateFamily: cirq.ops.common_gates.XPowGate",
+        "description": "Accepts `cirq.Gate` instances `g` s.t. `isinstance(g, cirq.ops.common_gates.XPowGate)`",
+        "ignore_global_phase": true,
+        "tags_to_accept": [],
+        "tags_to_ignore": []
+      },
+      {
+        "cirq_type": "GateFamily",
+        "gate": "ZPowGate",
+        "name": "Type GateFamily: cirq.ops.common_gates.ZPowGate",
+        "description": "Accepts `cirq.Gate` instances `g` s.t. `isinstance(g, cirq.ops.common_gates.ZPowGate)`",
+        "ignore_global_phase": true,
+        "tags_to_accept": [],
+        "tags_to_ignore": []
+      }
+    ]
   }
 ]
\ No newline at end of file
diff --git a/cirq-core/cirq/protocols/json_test_data/CZTargetGateset.repr b/cirq-core/cirq/protocols/json_test_data/CZTargetGateset.repr
index 0821e1394d6..3d0df0caa6f 100644
--- a/cirq-core/cirq/protocols/json_test_data/CZTargetGateset.repr
+++ b/cirq-core/cirq/protocols/json_test_data/CZTargetGateset.repr
@@ -1,4 +1,18 @@
 [
-    cirq.CZTargetGateset(atol=1e-06, allow_partial_czs=False),
-    cirq.CZTargetGateset(atol=1e-08, allow_partial_czs=True),
+    cirq.CZTargetGateset(atol=1e-06, allow_partial_czs=False, additional_gates=[]),
+    cirq.CZTargetGateset(atol=1e-08, allow_partial_czs=True, additional_gates=[]),
+    cirq.CZTargetGateset(
+        atol=1e-06,
+        allow_partial_czs=True,
+        additional_gates=[
+            (cirq.ISWAP**0.5),
+            cirq.ops.common_gates.XPowGate,
+            cirq.GateFamily(
+                gate=cirq.ops.common_gates.ZPowGate,
+                ignore_global_phase=True,
+                tags_to_accept=frozenset(),
+                tags_to_ignore=frozenset(),
+            ),
+        ],
+    ),
 ]
diff --git a/cirq-core/cirq/protocols/json_test_data/SqrtIswapTargetGateset.json b/cirq-core/cirq/protocols/json_test_data/SqrtIswapTargetGateset.json
index d52e44147c4..31b16b6b82c 100644
--- a/cirq-core/cirq/protocols/json_test_data/SqrtIswapTargetGateset.json
+++ b/cirq-core/cirq/protocols/json_test_data/SqrtIswapTargetGateset.json
@@ -16,5 +16,44 @@
     "atol": 1e-06,
     "required_sqrt_iswap_count": 2,
     "use_sqrt_iswap_inv": true
+  },
+  {
+    "cirq_type": "SqrtIswapTargetGateset",
+    "atol": 1e-08,
+    "required_sqrt_iswap_count": null,
+    "use_sqrt_iswap_inv": false,
+    "additional_gates": [
+      {
+        "cirq_type": "GateFamily",
+        "gate": {
+          "cirq_type": "CZPowGate",
+          "exponent": 1.0,
+          "global_shift": 0.0
+        },
+        "name": "Instance GateFamily: CZ",
+        "description": "Accepts `cirq.Gate` instances `g` s.t. `g == CZ`",
+        "ignore_global_phase": true,
+        "tags_to_accept": [],
+        "tags_to_ignore": []
+      },
+      {
+        "cirq_type": "GateFamily",
+        "gate": "XPowGate",
+        "name": "Type GateFamily: cirq.ops.common_gates.XPowGate",
+        "description": "Accepts `cirq.Gate` instances `g` s.t. `isinstance(g, cirq.ops.common_gates.XPowGate)`",
+        "ignore_global_phase": true,
+        "tags_to_accept": [],
+        "tags_to_ignore": []
+      },
+      {
+        "cirq_type": "GateFamily",
+        "gate": "ZPowGate",
+        "name": "Type GateFamily: cirq.ops.common_gates.ZPowGate",
+        "description": "Accepts `cirq.Gate` instances `g` s.t. `isinstance(g, cirq.ops.common_gates.ZPowGate)`",
+        "ignore_global_phase": true,
+        "tags_to_accept": [],
+        "tags_to_ignore": []
+      }
+    ]
   }
 ]
\ No newline at end of file
diff --git a/cirq-core/cirq/protocols/json_test_data/SqrtIswapTargetGateset.repr b/cirq-core/cirq/protocols/json_test_data/SqrtIswapTargetGateset.repr
index 06c6315ca94..09527eb9516 100644
--- a/cirq-core/cirq/protocols/json_test_data/SqrtIswapTargetGateset.repr
+++ b/cirq-core/cirq/protocols/json_test_data/SqrtIswapTargetGateset.repr
@@ -1,7 +1,26 @@
 [
     cirq.SqrtIswapTargetGateset(
-        atol=1e-08, required_sqrt_iswap_count=None, use_sqrt_iswap_inv=False
+        atol=1e-08, required_sqrt_iswap_count=None, use_sqrt_iswap_inv=False, additional_gates=[]
+    ),
+    cirq.SqrtIswapTargetGateset(
+        atol=1e-08, required_sqrt_iswap_count=1, use_sqrt_iswap_inv=False, additional_gates=[]
+    ),
+    cirq.SqrtIswapTargetGateset(
+        atol=1e-06, required_sqrt_iswap_count=2, use_sqrt_iswap_inv=True, additional_gates=[]
+    ),
+    cirq.SqrtIswapTargetGateset(
+        atol=1e-08,
+        required_sqrt_iswap_count=None,
+        use_sqrt_iswap_inv=False,
+        additional_gates=[
+            cirq.CZ,
+            cirq.ops.common_gates.XPowGate,
+            cirq.GateFamily(
+                gate=cirq.ops.common_gates.ZPowGate,
+                ignore_global_phase=True,
+                tags_to_accept=frozenset(),
+                tags_to_ignore=frozenset(),
+            ),
+        ],
     ),
-    cirq.SqrtIswapTargetGateset(atol=1e-08, required_sqrt_iswap_count=1, use_sqrt_iswap_inv=False),
-    cirq.SqrtIswapTargetGateset(atol=1e-06, required_sqrt_iswap_count=2, use_sqrt_iswap_inv=True),
 ]
diff --git a/cirq-core/cirq/transformers/target_gatesets/cz_gateset.py b/cirq-core/cirq/transformers/target_gatesets/cz_gateset.py
index fa83f481120..af00f242a9d 100644
--- a/cirq-core/cirq/transformers/target_gatesets/cz_gateset.py
+++ b/cirq-core/cirq/transformers/target_gatesets/cz_gateset.py
@@ -14,7 +14,7 @@
 
 """Target gateset used for compiling circuits to CZ + 1-q rotations + measurement gates."""
 
-from typing import Any, Dict, TYPE_CHECKING
+from typing import Any, Dict, Sequence, Type, Union, TYPE_CHECKING
 
 from cirq import ops, protocols
 from cirq.transformers.analytical_decompositions import two_qubit_to_cz
@@ -25,23 +25,53 @@
 
 
 class CZTargetGateset(compilation_target_gateset.TwoQubitCompilationTargetGateset):
-    """Target gateset containing CZ + single qubit rotations + Measurement gates."""
+    """Target gateset accepting CZ + single qubit rotations + measurement gates.
 
-    def __init__(self, *, atol: float = 1e-8, allow_partial_czs: bool = False) -> None:
+    By default, `cirq.CZTargetGateset` will accept and compile unknown gates to
+    the following universal target gateset:
+    - `cirq.CZ` / `cirq.CZPowGate`: The two qubit entangling gate.
+    - `cirq.PhasedXZGate`: Single qubit rotations.
+    - `cirq.MeasurementGate`: Measurements.
+    - `cirq.GlobalPhaseGate`: Global phase.
+
+    Optionally, users can also specify additional gates / gate families which should
+    be accepted by this gateset via the `additional_gates` argument.
+
+    When compiling a circuit, any unknown gate, i.e. a gate which is not accepted by
+    this gateset, will be compiled to the default gateset (i.e. `cirq.CZ`/`cirq.CZPowGate`,
+    `cirq.PhasedXZGate`, `cirq.MeasurementGate`).
+    """
+
+    def __init__(
+        self,
+        *,
+        atol: float = 1e-8,
+        allow_partial_czs: bool = False,
+        additional_gates: Sequence[Union[Type['cirq.Gate'], 'cirq.Gate', 'cirq.GateFamily']] = (),
+    ) -> None:
         """Initializes CZTargetGateset
 
         Args:
             atol: A limit on the amount of absolute error introduced by the decomposition.
             allow_partial_czs: If set, all powers of the form `cirq.CZ**t`, and not just
              `cirq.CZ`, are part of this gateset.
+            additional_gates: Sequence of additional gates / gate families which should also
+              be "accepted" by this gateset. Defaults to `cirq.GlobalPhaseGate`.
         """
         super().__init__(
             ops.CZPowGate if allow_partial_czs else ops.CZ,
             ops.MeasurementGate,
-            ops.AnyUnitaryGateFamily(1),
+            ops.PhasedXZGate,
             ops.GlobalPhaseGate,
+            *additional_gates,
             name='CZPowTargetGateset' if allow_partial_czs else 'CZTargetGateset',
         )
+        self.additional_gates = tuple(
+            g if isinstance(g, ops.GateFamily) else ops.GateFamily(gate=g) for g in additional_gates
+        )
+        self._additional_gates_repr_str = ", ".join(
+            [ops.gateset._gate_str(g, repr) for g in additional_gates]
+        )
         self.atol = atol
         self.allow_partial_czs = allow_partial_czs
 
@@ -57,14 +87,25 @@ def _decompose_two_qubit_operation(self, op: 'cirq.Operation', _) -> 'cirq.OP_TR
         )
 
     def __repr__(self) -> str:
-        return f'cirq.CZTargetGateset(atol={self.atol}, allow_partial_czs={self.allow_partial_czs})'
+        return (
+            f'cirq.CZTargetGateset('
+            f'atol={self.atol}, '
+            f'allow_partial_czs={self.allow_partial_czs}, '
+            f'additional_gates=[{self._additional_gates_repr_str}]'
+            f')'
+        )
 
     def _value_equality_values_(self) -> Any:
-        return self.atol, self.allow_partial_czs
+        return self.atol, self.allow_partial_czs, frozenset(self.additional_gates)
 
     def _json_dict_(self) -> Dict[str, Any]:
-        return {'atol': self.atol, 'allow_partial_czs': self.allow_partial_czs}
+        d: Dict[str, Any] = {'atol': self.atol, 'allow_partial_czs': self.allow_partial_czs}
+        if self.additional_gates:
+            d['additional_gates'] = list(self.additional_gates)
+        return d
 
     @classmethod
-    def _from_json_dict_(cls, atol, allow_partial_czs, **kwargs):
-        return cls(atol=atol, allow_partial_czs=allow_partial_czs)
+    def _from_json_dict_(cls, atol, allow_partial_czs, additional_gates=(), **kwargs):
+        return cls(
+            atol=atol, allow_partial_czs=allow_partial_czs, additional_gates=additional_gates
+        )
diff --git a/cirq-core/cirq/transformers/target_gatesets/cz_gateset_test.py b/cirq-core/cirq/transformers/target_gatesets/cz_gateset_test.py
index 40952cd895b..00f31414d2a 100644
--- a/cirq-core/cirq/transformers/target_gatesets/cz_gateset_test.py
+++ b/cirq-core/cirq/transformers/target_gatesets/cz_gateset_test.py
@@ -12,6 +12,7 @@
 # See the License for the specific language governing permissions and
 # limitations under the License.
 
+from typing import Optional, Sequence, Type
 import pytest
 import cirq
 import sympy
@@ -25,9 +26,18 @@ def all_gates_of_type(m: cirq.Moment, g: cirq.Gateset):
     return True
 
 
-def assert_optimizes(before: cirq.Circuit, expected: cirq.Circuit):
+def assert_optimizes(
+    before: cirq.Circuit,
+    expected: cirq.Circuit,
+    additional_gates: Optional[Sequence[Type[cirq.Gate]]] = None,
+):
+    if additional_gates is None:
+        gateset = cirq.CZTargetGateset()
+    else:
+        gateset = cirq.CZTargetGateset(additional_gates=additional_gates)
+
     cirq.testing.assert_same_circuits(
-        cirq.optimize_for_target_gateset(before, gateset=cirq.CZTargetGateset()), expected
+        cirq.optimize_for_target_gateset(before, gateset=gateset, ignore_failures=False), expected
     )
 
 
@@ -37,7 +47,7 @@ def assert_optimization_not_broken(circuit: cirq.Circuit):
         circuit, c_new, atol=1e-6
     )
     c_new = cirq.optimize_for_target_gateset(
-        circuit, gateset=cirq.CZTargetGateset(allow_partial_czs=True)
+        circuit, gateset=cirq.CZTargetGateset(allow_partial_czs=True), ignore_failures=False
     )
     cirq.testing.assert_circuits_with_terminal_measurements_are_equivalent(
         circuit, c_new, atol=1e-6
@@ -57,7 +67,11 @@ def test_convert_to_cz_preserving_moment_structure():
         cirq.X(q[2]).with_classical_controls("m"),
         cirq.CZ(*q[3:]).with_classical_controls("m"),
     )
-    c_new = cirq.optimize_for_target_gateset(c_orig, gateset=cirq.CZTargetGateset())
+    # Classically controlled operations are not part of the gateset, so failures should be ignored
+    # during compilation.
+    c_new = cirq.optimize_for_target_gateset(
+        c_orig, gateset=cirq.CZTargetGateset(), ignore_failures=True
+    )
 
     assert c_orig[-2:] == c_new[-2:]
     c_orig, c_new = c_orig[:-2], c_new[:-2]
@@ -65,7 +79,7 @@ def test_convert_to_cz_preserving_moment_structure():
     cirq.testing.assert_circuits_with_terminal_measurements_are_equivalent(c_orig, c_new, atol=1e-6)
     assert all(
         (
-            all_gates_of_type(m, cirq.Gateset(cirq.AnyUnitaryGateFamily(1)))
+            all_gates_of_type(m, cirq.Gateset(cirq.PhasedXZGate))
             or all_gates_of_type(m, cirq.Gateset(cirq.CZ))
         )
         for m in c_new
@@ -77,7 +91,7 @@ def test_convert_to_cz_preserving_moment_structure():
     cirq.testing.assert_circuits_with_terminal_measurements_are_equivalent(c_orig, c_new, atol=1e-6)
     assert all(
         (
-            all_gates_of_type(m, cirq.Gateset(cirq.AnyUnitaryGateFamily(1)))
+            all_gates_of_type(m, cirq.Gateset(cirq.PhasedXZGate))
             or all_gates_of_type(m, cirq.Gateset(cirq.CZPowGate))
         )
         for m in c_new
@@ -109,6 +123,7 @@ def test_ignores_czs_separated_by_parameterized():
                 cirq.Moment(cirq.CZ(a, b)),
             ]
         ),
+        additional_gates=[cirq.ZPowGate],
     )
 
 
@@ -153,7 +168,7 @@ def test_optimizes_single_iswap():
     a, b = cirq.LineQubit.range(2)
     c = cirq.Circuit(cirq.ISWAP(a, b))
     assert_optimization_not_broken(c)
-    c = cirq.optimize_for_target_gateset(c, gateset=cirq.CZTargetGateset())
+    c = cirq.optimize_for_target_gateset(c, gateset=cirq.CZTargetGateset(), ignore_failures=False)
     assert len([1 for op in c.all_operations() if len(op.qubits) == 2]) == 2
 
 
@@ -161,7 +176,7 @@ def test_optimizes_tagged_partial_cz():
     a, b = cirq.LineQubit.range(2)
     c = cirq.Circuit((cirq.CZ**0.5)(a, b).with_tags('mytag'))
     assert_optimization_not_broken(c)
-    c = cirq.optimize_for_target_gateset(c, gateset=cirq.CZTargetGateset())
+    c = cirq.optimize_for_target_gateset(c, gateset=cirq.CZTargetGateset(), ignore_failures=False)
     assert (
         len([1 for op in c.all_operations() if len(op.qubits) == 2]) == 2
     ), 'It should take 2 CZ gates to decompose a CZ**0.5 gate'
@@ -185,7 +200,9 @@ def test_not_decompose_czs():
     ),
 )
 def test_decompose_partial_czs(circuit):
-    circuit = cirq.optimize_for_target_gateset(circuit, gateset=cirq.CZTargetGateset())
+    circuit = cirq.optimize_for_target_gateset(
+        circuit, gateset=cirq.CZTargetGateset(), ignore_failures=False
+    )
     cz_gates = [
         op.gate
         for op in circuit.all_operations()
@@ -201,7 +218,7 @@ def test_not_decompose_partial_czs():
     circuit = cirq.Circuit(
         cirq.CZPowGate(exponent=0.1, global_shift=-0.5)(*cirq.LineQubit.range(2))
     )
-    cirq.optimize_for_target_gateset(circuit, gateset=cirq.CZTargetGateset())
+    cirq.optimize_for_target_gateset(circuit, gateset=cirq.CZTargetGateset(), ignore_failures=False)
     cz_gates = [
         op.gate
         for op in circuit.all_operations()
@@ -240,7 +257,7 @@ def _decompose_(self, qubits):
 
     a, b = cirq.LineQubit.range(2)
     c = cirq.Circuit(OtherXX()(a, b), OtherOtherXX()(a, b))
-    c = cirq.optimize_for_target_gateset(c, gateset=cirq.CZTargetGateset())
+    c = cirq.optimize_for_target_gateset(c, gateset=cirq.CZTargetGateset(), ignore_failures=False)
     assert len(c) == 0
 
 
@@ -260,7 +277,9 @@ def _decompose_(self, qubits):
     expected = cirq.Circuit(
         cirq.X(q0), cirq.Y(q0) ** 0.5, cirq.CZ(q0, q1), cirq.X(q1), cirq.Y(q1) ** 0.5
     )
-    c_new = cirq.optimize_for_target_gateset(circuit, gateset=cirq.CZTargetGateset())
+    c_new = cirq.optimize_for_target_gateset(
+        circuit, gateset=cirq.CZTargetGateset(), ignore_failures=False
+    )
 
     cirq.testing.assert_circuits_with_terminal_measurements_are_equivalent(
         c_new, expected, atol=1e-6
@@ -281,3 +300,24 @@ class UnsupportedDummy(cirq.testing.TwoQubitGate):
         _ = cirq.optimize_for_target_gateset(
             circuit, gateset=cirq.CZTargetGateset(), ignore_failures=False
         )
+
+
+@pytest.mark.parametrize(
+    'gateset',
+    [
+        cirq.CZTargetGateset(),
+        cirq.CZTargetGateset(
+            atol=1e-6,
+            allow_partial_czs=True,
+            additional_gates=[
+                cirq.SQRT_ISWAP,
+                cirq.XPowGate,
+                cirq.YPowGate,
+                cirq.GateFamily(cirq.ZPowGate, tags_to_accept=['test_tag']),
+            ],
+        ),
+        cirq.CZTargetGateset(additional_gates=()),
+    ],
+)
+def test_repr(gateset):
+    cirq.testing.assert_equivalent_repr(gateset)
diff --git a/cirq-core/cirq/transformers/target_gatesets/sqrt_iswap_gateset.py b/cirq-core/cirq/transformers/target_gatesets/sqrt_iswap_gateset.py
index f6fa3f93c90..5d26665d599 100644
--- a/cirq-core/cirq/transformers/target_gatesets/sqrt_iswap_gateset.py
+++ b/cirq-core/cirq/transformers/target_gatesets/sqrt_iswap_gateset.py
@@ -14,7 +14,7 @@
 
 """Target gateset used for compiling circuits to √iSWAP + 1-q rotations + measurement gates."""
 
-from typing import Any, Dict, Optional, TYPE_CHECKING
+from typing import Any, Dict, Optional, Sequence, Type, Union, TYPE_CHECKING
 
 from cirq import ops, protocols
 from cirq.protocols.decompose_protocol import DecomposeResult
@@ -26,7 +26,22 @@
 
 
 class SqrtIswapTargetGateset(compilation_target_gateset.TwoQubitCompilationTargetGateset):
-    """Target gateset containing √iSWAP + single qubit rotations + Measurement gates."""
+    """Target gateset accepting √iSWAP + single qubit rotations + measurement gates.
+
+    By default, `cirq.SqrtIswapTargetGateset` will accept and compile unknown gates to
+    the following universal target gateset:
+    - `cirq.SQRT_ISWAP` / `cirq.SQRT_ISWAP_INV`: The two qubit entangling gate.
+    - `cirq.PhasedXZGate`: Single qubit rotations.
+    - `cirq.MeasurementGate`: Measurements.
+    - `cirq.GlobalPhaseGate`: Global phase.
+
+    Optionally, users can also specify additional gates / gate families which should
+    be accepted by this gateset via the `additional_gates` argument.
+
+    When compiling a circuit, any unknown gate, i.e. a gate which is not accepted by
+    this gateset, will be compiled to the default gateset (i.e. `cirq.SQRT_ISWAP`/
+    `cirq.cirq.SQRT_ISWAP_INV`, `cirq.PhasedXZGate`, `cirq.MeasurementGate`).
+    """
 
     def __init__(
         self,
@@ -34,6 +49,7 @@ def __init__(
         atol: float = 1e-8,
         required_sqrt_iswap_count: Optional[int] = None,
         use_sqrt_iswap_inv: bool = False,
+        additional_gates: Sequence[Union[Type['cirq.Gate'], 'cirq.Gate', 'cirq.GateFamily']] = (),
     ):
         """Initializes `cirq.SqrtIswapTargetGateset`
 
@@ -45,6 +61,8 @@ def __init__(
                 synthesis of the operation requires more.
             use_sqrt_iswap_inv: If True, `cirq.SQRT_ISWAP_INV` is used as part of the gateset,
                 instead of `cirq.SQRT_ISWAP`.
+            additional_gates: Sequence of additional gates / gate families which should also
+              be "accepted" by this gateset. Defaults to `cirq.GlobalPhaseGate`.
 
         Raises:
             ValueError: If `required_sqrt_iswap_count` is specified and is not 0, 1, 2, or 3.
@@ -54,10 +72,17 @@ def __init__(
         super().__init__(
             ops.SQRT_ISWAP_INV if use_sqrt_iswap_inv else ops.SQRT_ISWAP,
             ops.MeasurementGate,
-            ops.AnyUnitaryGateFamily(1),
+            ops.PhasedXZGate,
             ops.GlobalPhaseGate,
+            *additional_gates,
             name='SqrtIswapInvTargetGateset' if use_sqrt_iswap_inv else 'SqrtIswapTargetGateset',
         )
+        self.additional_gates = tuple(
+            g if isinstance(g, ops.GateFamily) else ops.GateFamily(gate=g) for g in additional_gates
+        )
+        self._additional_gates_repr_str = ", ".join(
+            [ops.gateset._gate_str(g, repr) for g in additional_gates]
+        )
         self.atol = atol
         self.required_sqrt_iswap_count = required_sqrt_iswap_count
         self.use_sqrt_iswap_inv = use_sqrt_iswap_inv
@@ -85,24 +110,36 @@ def __repr__(self) -> str:
             f'cirq.SqrtIswapTargetGateset('
             f'atol={self.atol}, '
             f'required_sqrt_iswap_count={self.required_sqrt_iswap_count}, '
-            f'use_sqrt_iswap_inv={self.use_sqrt_iswap_inv}'
+            f'use_sqrt_iswap_inv={self.use_sqrt_iswap_inv}, '
+            f'additional_gates=[{self._additional_gates_repr_str}]'
             f')'
         )
 
     def _value_equality_values_(self) -> Any:
-        return (self.atol, self.required_sqrt_iswap_count, self.use_sqrt_iswap_inv)
+        return (
+            self.atol,
+            self.required_sqrt_iswap_count,
+            self.use_sqrt_iswap_inv,
+            frozenset(self.additional_gates),
+        )
 
     def _json_dict_(self) -> Dict[str, Any]:
-        return {
+        d: Dict[str, Any] = {
             'atol': self.atol,
             'required_sqrt_iswap_count': self.required_sqrt_iswap_count,
             'use_sqrt_iswap_inv': self.use_sqrt_iswap_inv,
         }
+        if self.additional_gates:
+            d['additional_gates'] = list(self.additional_gates)
+        return d
 
     @classmethod
-    def _from_json_dict_(cls, atol, required_sqrt_iswap_count, use_sqrt_iswap_inv, **kwargs):
+    def _from_json_dict_(
+        cls, atol, required_sqrt_iswap_count, use_sqrt_iswap_inv, additional_gates=(), **kwargs
+    ):
         return cls(
             atol=atol,
             required_sqrt_iswap_count=required_sqrt_iswap_count,
             use_sqrt_iswap_inv=use_sqrt_iswap_inv,
+            additional_gates=additional_gates,
         )
diff --git a/cirq-core/cirq/transformers/target_gatesets/sqrt_iswap_gateset_test.py b/cirq-core/cirq/transformers/target_gatesets/sqrt_iswap_gateset_test.py
index 9f2af0fd3ba..b2cf7ed4076 100644
--- a/cirq-core/cirq/transformers/target_gatesets/sqrt_iswap_gateset_test.py
+++ b/cirq-core/cirq/transformers/target_gatesets/sqrt_iswap_gateset_test.py
@@ -29,7 +29,9 @@ def all_gates_of_type(m: cirq.Moment, g: cirq.Gateset):
 
 def assert_optimizes(before: cirq.Circuit, expected: cirq.Circuit, **kwargs):
     cirq.testing.assert_same_circuits(
-        cirq.optimize_for_target_gateset(before, gateset=cirq.SqrtIswapTargetGateset(**kwargs)),
+        cirq.optimize_for_target_gateset(
+            before, gateset=cirq.SqrtIswapTargetGateset(**kwargs), ignore_failures=False
+        ),
         expected,
     )
 
@@ -40,6 +42,7 @@ def assert_optimization_not_broken(
     c_new = cirq.optimize_for_target_gateset(
         circuit,
         gateset=cirq.SqrtIswapTargetGateset(required_sqrt_iswap_count=required_sqrt_iswap_count),
+        ignore_failures=False,
     )
     cirq.testing.assert_circuits_with_terminal_measurements_are_equivalent(
         circuit, c_new, atol=1e-6
@@ -49,6 +52,7 @@ def assert_optimization_not_broken(
         gateset=cirq.SqrtIswapTargetGateset(
             use_sqrt_iswap_inv=True, required_sqrt_iswap_count=required_sqrt_iswap_count
         ),
+        ignore_failures=False,
     )
     cirq.testing.assert_circuits_with_terminal_measurements_are_equivalent(
         circuit, c_new, atol=1e-6
@@ -68,8 +72,11 @@ def test_convert_to_sqrt_iswap_preserving_moment_structure():
         cirq.X(q[2]).with_classical_controls("m"),
         cirq.CZ(*q[3:]).with_classical_controls("m"),
     )
-
-    c_new = cirq.optimize_for_target_gateset(c_orig, gateset=cirq.SqrtIswapTargetGateset())
+    # Classically controlled operations are not part of the gateset, so failures should be ignored
+    # during compilation.
+    c_new = cirq.optimize_for_target_gateset(
+        c_orig, gateset=cirq.SqrtIswapTargetGateset(), ignore_failures=True
+    )
 
     assert c_orig[-2:] == c_new[-2:]
     c_orig, c_new = c_orig[:-2], c_new[:-2]
@@ -77,7 +84,7 @@ def test_convert_to_sqrt_iswap_preserving_moment_structure():
     cirq.testing.assert_circuits_with_terminal_measurements_are_equivalent(c_orig, c_new, atol=1e-6)
     assert all(
         (
-            all_gates_of_type(m, cirq.Gateset(cirq.AnyUnitaryGateFamily(1)))
+            all_gates_of_type(m, cirq.Gateset(cirq.PhasedXZGate))
             or all_gates_of_type(m, cirq.Gateset(cirq.SQRT_ISWAP))
         )
         for m in c_new
@@ -89,7 +96,7 @@ def test_convert_to_sqrt_iswap_preserving_moment_structure():
     cirq.testing.assert_circuits_with_terminal_measurements_are_equivalent(c_orig, c_new, atol=1e-6)
     assert all(
         (
-            all_gates_of_type(m, cirq.Gateset(cirq.AnyUnitaryGateFamily(1)))
+            all_gates_of_type(m, cirq.Gateset(cirq.PhasedXZGate))
             or all_gates_of_type(m, cirq.Gateset(cirq.SQRT_ISWAP_INV))
         )
         for m in c_new
@@ -111,7 +118,12 @@ def test_two_qubit_gates_with_symbols(gate: cirq.Gate, use_sqrt_iswap_inv: bool)
 
     c_orig = cirq.Circuit(gate(*cirq.LineQubit.range(2)))
     c_new = cirq.optimize_for_target_gateset(
-        c_orig, gateset=cirq.SqrtIswapTargetGateset(use_sqrt_iswap_inv=use_sqrt_iswap_inv)
+        c_orig,
+        gateset=cirq.SqrtIswapTargetGateset(
+            use_sqrt_iswap_inv=use_sqrt_iswap_inv,
+            additional_gates=[cirq.XPowGate, cirq.YPowGate, cirq.ZPowGate],
+        ),
+        ignore_failures=False,
     )
 
     # Check that `c_new` only contains sqrt iswap as the 2q entangling gate.
@@ -145,6 +157,7 @@ def test_sqrt_iswap_gateset_eq():
     eq.add_equality_group(
         cirq.SqrtIswapTargetGateset(atol=1e-6, required_sqrt_iswap_count=3, use_sqrt_iswap_inv=True)
     )
+    eq.add_equality_group(cirq.SqrtIswapTargetGateset(additional_gates=[cirq.XPowGate]))
 
 
 @pytest.mark.parametrize(
@@ -152,8 +165,17 @@ def test_sqrt_iswap_gateset_eq():
     [
         cirq.SqrtIswapTargetGateset(),
         cirq.SqrtIswapTargetGateset(
-            atol=1e-6, required_sqrt_iswap_count=2, use_sqrt_iswap_inv=True
+            atol=1e-6,
+            required_sqrt_iswap_count=2,
+            use_sqrt_iswap_inv=True,
+            additional_gates=[
+                cirq.CZ,
+                cirq.XPowGate,
+                cirq.YPowGate,
+                cirq.GateFamily(cirq.ZPowGate, tags_to_accept=['test_tag']),
+            ],
         ),
+        cirq.SqrtIswapTargetGateset(additional_gates=()),
     ],
 )
 def test_sqrt_iswap_gateset_repr(gateset):
@@ -282,7 +304,9 @@ def test_optimizes_single_iswap():
     a, b = cirq.LineQubit.range(2)
     c = cirq.Circuit(cirq.ISWAP(a, b))
     assert_optimization_not_broken(c)
-    c = cirq.optimize_for_target_gateset(c, gateset=cirq.SqrtIswapTargetGateset())
+    c = cirq.optimize_for_target_gateset(
+        c, gateset=cirq.SqrtIswapTargetGateset(), ignore_failures=False
+    )
     assert len([1 for op in c.all_operations() if len(op.qubits) == 2]) == 2
 
 
@@ -290,7 +314,9 @@ def test_optimizes_single_inv_sqrt_iswap():
     a, b = cirq.LineQubit.range(2)
     c = cirq.Circuit(cirq.SQRT_ISWAP_INV(a, b))
     assert_optimization_not_broken(c)
-    c = cirq.optimize_for_target_gateset(c, gateset=cirq.SqrtIswapTargetGateset())
+    c = cirq.optimize_for_target_gateset(
+        c, gateset=cirq.SqrtIswapTargetGateset(), ignore_failures=False
+    )
     assert len([1 for op in c.all_operations() if len(op.qubits) == 2]) == 1
 
 
@@ -299,7 +325,7 @@ def test_optimizes_single_iswap_require0():
     c = cirq.Circuit(cirq.CNOT(a, b), cirq.CNOT(a, b))  # Minimum 0 sqrt-iSWAP
     assert_optimization_not_broken(c, required_sqrt_iswap_count=0)
     c = cirq.optimize_for_target_gateset(
-        c, gateset=cirq.SqrtIswapTargetGateset(required_sqrt_iswap_count=0)
+        c, gateset=cirq.SqrtIswapTargetGateset(required_sqrt_iswap_count=0), ignore_failures=False
     )
     assert len([1 for op in c.all_operations() if len(op.qubits) == 2]) == 0
 
@@ -309,7 +335,9 @@ def test_optimizes_single_iswap_require0_raises():
     c = cirq.Circuit(cirq.CNOT(a, b))  # Minimum 2 sqrt-iSWAP
     with pytest.raises(ValueError, match='cannot be decomposed into exactly 0 sqrt-iSWAP gates'):
         _ = cirq.optimize_for_target_gateset(
-            c, gateset=cirq.SqrtIswapTargetGateset(required_sqrt_iswap_count=0)
+            c,
+            gateset=cirq.SqrtIswapTargetGateset(required_sqrt_iswap_count=0),
+            ignore_failures=False,
         )
 
 
@@ -318,7 +346,7 @@ def test_optimizes_single_iswap_require1():
     c = cirq.Circuit(cirq.SQRT_ISWAP_INV(a, b))  # Minimum 1 sqrt-iSWAP
     assert_optimization_not_broken(c, required_sqrt_iswap_count=1)
     c = cirq.optimize_for_target_gateset(
-        c, gateset=cirq.SqrtIswapTargetGateset(required_sqrt_iswap_count=1)
+        c, gateset=cirq.SqrtIswapTargetGateset(required_sqrt_iswap_count=1), ignore_failures=False
     )
     assert len([1 for op in c.all_operations() if len(op.qubits) == 2]) == 1
 
@@ -328,7 +356,9 @@ def test_optimizes_single_iswap_require1_raises():
     c = cirq.Circuit(cirq.CNOT(a, b))  # Minimum 2 sqrt-iSWAP
     with pytest.raises(ValueError, match='cannot be decomposed into exactly 1 sqrt-iSWAP gates'):
         c = cirq.optimize_for_target_gateset(
-            c, gateset=cirq.SqrtIswapTargetGateset(required_sqrt_iswap_count=1)
+            c,
+            gateset=cirq.SqrtIswapTargetGateset(required_sqrt_iswap_count=1),
+            ignore_failures=False,
         )
 
 
@@ -337,7 +367,7 @@ def test_optimizes_single_iswap_require2():
     c = cirq.Circuit(cirq.SQRT_ISWAP_INV(a, b))  # Minimum 1 sqrt-iSWAP but 2 possible
     assert_optimization_not_broken(c, required_sqrt_iswap_count=2)
     c = cirq.optimize_for_target_gateset(
-        c, gateset=cirq.SqrtIswapTargetGateset(required_sqrt_iswap_count=2)
+        c, gateset=cirq.SqrtIswapTargetGateset(required_sqrt_iswap_count=2), ignore_failures=False
     )
     assert len([1 for op in c.all_operations() if len(op.qubits) == 2]) == 2
 
@@ -347,7 +377,9 @@ def test_optimizes_single_iswap_require2_raises():
     c = cirq.Circuit(cirq.SWAP(a, b))  # Minimum 3 sqrt-iSWAP
     with pytest.raises(ValueError, match='cannot be decomposed into exactly 2 sqrt-iSWAP gates'):
         c = cirq.optimize_for_target_gateset(
-            c, gateset=cirq.SqrtIswapTargetGateset(required_sqrt_iswap_count=2)
+            c,
+            gateset=cirq.SqrtIswapTargetGateset(required_sqrt_iswap_count=2),
+            ignore_failures=False,
         )
 
 
@@ -356,7 +388,7 @@ def test_optimizes_single_iswap_require3():
     c = cirq.Circuit(cirq.ISWAP(a, b))  # Minimum 2 sqrt-iSWAP but 3 possible
     assert_optimization_not_broken(c, required_sqrt_iswap_count=3)
     c = cirq.optimize_for_target_gateset(
-        c, gateset=cirq.SqrtIswapTargetGateset(required_sqrt_iswap_count=3)
+        c, gateset=cirq.SqrtIswapTargetGateset(required_sqrt_iswap_count=3), ignore_failures=False
     )
     assert len([1 for op in c.all_operations() if len(op.qubits) == 2]) == 3
 
@@ -366,6 +398,6 @@ def test_optimizes_single_inv_sqrt_iswap_require3():
     c = cirq.Circuit(cirq.SQRT_ISWAP_INV(a, b))
     assert_optimization_not_broken(c, required_sqrt_iswap_count=3)
     c = cirq.optimize_for_target_gateset(
-        c, gateset=cirq.SqrtIswapTargetGateset(required_sqrt_iswap_count=3)
+        c, gateset=cirq.SqrtIswapTargetGateset(required_sqrt_iswap_count=3), ignore_failures=False
     )
     assert len([1 for op in c.all_operations() if len(op.qubits) == 2]) == 3