diff --git a/ibis/backends/tests/test_impure.py b/ibis/backends/tests/test_impure.py
new file mode 100644
index 0000000000000..648e2cd001b25
--- /dev/null
+++ b/ibis/backends/tests/test_impure.py
@@ -0,0 +1,149 @@
+from __future__ import annotations
+
+import pandas.testing as tm
+import pytest
+
+import ibis
+import ibis.common.exceptions as com
+from ibis import _
+from ibis.backends.tests.errors import (
+    PsycoPg2InternalError,
+    PyDruidProgrammingError,
+)
+
+no_random = [
+    pytest.mark.notimpl(
+        ["dask", "pandas", "polars"], raises=com.OperationNotDefinedError
+    ),
+    pytest.mark.notimpl(["druid"], raises=PyDruidProgrammingError),
+    pytest.mark.notimpl(
+        ["risingwave"],
+        raises=PsycoPg2InternalError,
+        reason="function random() does not exist",
+    ),
+]
+
+no_udfs = [
+    pytest.mark.notyet(["datafusion"], raises=NotImplementedError),
+    pytest.mark.notimpl(
+        [
+            "bigquery",
+            "clickhouse",
+            "dask",
+            "druid",
+            "exasol",
+            "impala",
+            "mssql",
+            "mysql",
+            "oracle",
+            "pandas",
+            "trino",
+            "risingwave",
+        ]
+    ),
+]
+
+no_uuid = pytest.mark.notimpl(
+    ["druid", "exasol", "oracle", "polars", "pyspark", "risingwave", "pandas", "dask"],
+    raises=com.OperationNotDefinedError,
+)
+
+
+@ibis.udf.scalar.python(side_effects=True)
+def my_random(x: float) -> float:
+    # need to make the whole UDF self-contained for postgres to work
+    import random
+
+    return random.random()
+
+
+mark_impures = pytest.mark.parametrize(
+    "impure",
+    [
+        pytest.param(
+            lambda _: ibis.random(),
+            marks=no_random,
+            id="random",
+        ),
+        pytest.param(
+            lambda _: ibis.uuid().cast(str).contains("a").ifelse(1, 0),
+            marks=no_uuid,
+            id="uuid",
+        ),
+        pytest.param(
+            lambda table: my_random(table.float_col),
+            marks=[
+                # once this is fixed, can we unify these params with the other params?
+                pytest.mark.broken(
+                    ["postgres" "flink"], reason="instances are uncorrelated"
+                ),
+                *no_udfs,
+            ],
+            id="udf",
+        ),
+    ],
+)
+
+
+@pytest.mark.broken("sqlite", reason="instances are uncorrelated")
+@mark_impures
+def test_impure_correlated(alltypes, impure):
+    df = (
+        alltypes.select(common=impure(alltypes))
+        .select(x=_.common, y=_.common)
+        .execute()
+    )
+    tm.assert_series_equal(df.x, df.y, check_names=False)
+
+
+@pytest.mark.broken("sqlite", reason="instances are uncorrelated")
+@mark_impures
+def test_chained_selections(alltypes, impure):
+    # https://github.com/ibis-project/ibis/issues/8921#issue-2234327722
+    t = alltypes.mutate(num=impure(alltypes))
+    t = t.mutate(isbig=(t.num > 0.5))
+    df = t.select("num", "isbig").execute()
+    df["expected"] = df.num > 0.5
+    tm.assert_series_equal(df.isbig, df.expected, check_names=False)
+
+
+@pytest.mark.broken(["clickhouse"], reason="instances are correlated")
+@pytest.mark.parametrize(
+    "impure",
+    [
+        pytest.param(
+            lambda _: ibis.random(),
+            marks=[
+                *no_random,
+                pytest.mark.broken(["trino"], reason="instances are correlated"),
+            ],
+            id="random",
+        ),
+        pytest.param(
+            # make this a float so we can compare to .5
+            lambda _: ibis.uuid().cast(str).contains("a").ifelse(1, 0),
+            marks=[
+                no_uuid,
+                pytest.mark.broken(["trino"], reason="instances are correlated"),
+            ],
+            id="uuid",
+        ),
+        pytest.param(
+            lambda table: my_random(table.float_col),
+            # once this is fixed, can we unify these params with the other params?
+            marks=[
+                *no_udfs,
+                pytest.mark.broken(["duckdb"], reason="instances are correlated"),
+            ],
+            id="udf",
+        ),
+    ],
+)
+def test_impure_uncorrelated(alltypes, impure):
+    df = alltypes.select(x=impure(alltypes), y=impure(alltypes)).execute()
+    assert (df.x == df.y).mean() < 1
+    # Even if the two expressions have the exact same ID, they should still be
+    # uncorrelated
+    common = impure(alltypes)
+    df = alltypes.select(x=common, y=common).execute()
+    assert (df.x == df.y).mean() < 1