diff --git a/build/pkgs/sagemath_objects/spkg-install b/build/pkgs/sagemath_objects/spkg-install
index b21c198671b..6cc85e07e55 100755
--- a/build/pkgs/sagemath_objects/spkg-install
+++ b/build/pkgs/sagemath_objects/spkg-install
@@ -25,5 +25,5 @@ wheel=$(cd "$DIST_DIR" && sdh_store_wheel . && echo $wheel)
 ls -l "$wheel"
 
 if [ "$SAGE_CHECK" != no ]; then
-    tox -r -v -e sagepython-norequirements-sagewheels-nopypi --installpkg $wheel
+    tox -r -v -e sagepython-sagewheels-nopypi-norequirements --installpkg $wheel
 fi
diff --git a/pkgs/sagemath-categories/tox.ini b/pkgs/sagemath-categories/tox.ini
index 85f7c5aa260..44ca511ac22 100644
--- a/pkgs/sagemath-categories/tox.ini
+++ b/pkgs/sagemath-categories/tox.ini
@@ -25,14 +25,15 @@ passenv =
                              PKG_CONFIG_PATH
     # Parallel build
                              SAGE_NUM_THREADS
-    # SAGE_VENV only for finding the wheels
-    sagewheels:              SAGE_VENV
+                             SAGE_NUM_THREADS_PARALLEL
+    # SAGE_VENV only for referring to the basepython or finding the wheels
+    sagepython, sagewheels:  SAGE_VENV
     # Location of the wheels
     sagewheels:              SAGE_SPKG_WHEELS
 
 setenv =
     # Sage scripts such as sage-runtests like to use $HOME/.sage
-    HOME={envdir}
+                             HOME={envdir}
     # We supply pip options by environment variables so that they
     # apply both to the installation of the dependencies and of the package
     sagewheels:              PIP_FIND_LINKS=file://{env:SAGE_SPKG_WHEELS:{env:SAGE_VENV:{toxinidir}/../../../../venv}/var/lib/sage/wheels}
@@ -51,7 +52,16 @@ commands =
     bash -c 'cd {temp_dir} && SAGE_SRC=$(python -c "from sage.env import SAGE_SRC; print(SAGE_SRC)") && sage-runtests --initial --environment=sage.all__sagemath_categories --optional=sage $SAGE_SRC/sage/structure  || echo "(lots of doctest failures are expected)"'
 
 [testenv:sagepython]
-passenv =
-    SAGE_VENV
+basepython = {env:SAGE_VENV}/bin/python3
+
+[testenv:sagepython-sagewheels-nopypi]
+basepython = {env:SAGE_VENV}/bin/python3
+
+[testenv:sagepython-sagewheels-nopypi-norequirements]
+basepython = {env:SAGE_VENV}/bin/python3
+
+[testenv:sagepython-sagewheels]
+basepython = {env:SAGE_VENV}/bin/python3
 
+[testenv:sagepython-norequirements]
 basepython = {env:SAGE_VENV}/bin/python3
diff --git a/pkgs/sagemath-environment/tox.ini b/pkgs/sagemath-environment/tox.ini
index ce1ddc3ed08..f4875e6228c 100644
--- a/pkgs/sagemath-environment/tox.ini
+++ b/pkgs/sagemath-environment/tox.ini
@@ -16,9 +16,18 @@ isolated_build = True
 deps =
     !norequirements: -rrequirements.txt
 
+passenv =
+    # Parallel build
+                             SAGE_NUM_THREADS
+                             SAGE_NUM_THREADS_PARALLEL
+    # SAGE_VENV only for referring to the basepython or finding the wheels
+    sagepython, sagewheels:  SAGE_VENV
+    # Location of the wheels
+    sagewheels:              SAGE_SPKG_WHEELS
+
 setenv =
     # Sage scripts such as sage-runtests like to use $HOME/.sage
-    HOME={envdir}
+                             HOME={envdir}
     # We supply pip options by environment variables so that they
     # apply both to the installation of the dependencies and of the package
     sagewheels:              PIP_FIND_LINKS=file://{env:SAGE_SPKG_WHEELS:{env:SAGE_VENV:{toxinidir}/../../../../venv}/var/lib/sage/wheels}
@@ -32,7 +41,16 @@ commands =
     {envpython} -c 'import sys; "" in sys.path and sys.path.remove(""); from sage.features.all import all_features; print(sorted(all_features(), key=lambda x: x.name)); import sage.misc.package'
 
 [testenv:sagepython]
-passenv =
-    SAGE_VENV
+basepython = {env:SAGE_VENV}/bin/python3
+
+[testenv:sagepython-sagewheels-nopypi]
+basepython = {env:SAGE_VENV}/bin/python3
+
+[testenv:sagepython-sagewheels-nopypi-norequirements]
+basepython = {env:SAGE_VENV}/bin/python3
+
+[testenv:sagepython-sagewheels]
+basepython = {env:SAGE_VENV}/bin/python3
 
+[testenv:sagepython-norequirements]
 basepython = {env:SAGE_VENV}/bin/python3
diff --git a/pkgs/sagemath-objects/tox.ini b/pkgs/sagemath-objects/tox.ini
index 4e93e572e1e..babc128ff0c 100644
--- a/pkgs/sagemath-objects/tox.ini
+++ b/pkgs/sagemath-objects/tox.ini
@@ -23,14 +23,15 @@ passenv =
                              PKG_CONFIG_PATH
     # Parallel build
                              SAGE_NUM_THREADS
-    # SAGE_VENV only for finding the wheels
-    sagewheels:              SAGE_VENV
+                             SAGE_NUM_THREADS_PARALLEL
+    # SAGE_VENV only for referring to the basepython or finding the wheels
+    sagepython, sagewheels:  SAGE_VENV
     # Location of the wheels
     sagewheels:              SAGE_SPKG_WHEELS
 
 setenv =
     # Sage scripts such as sage-runtests like to use $HOME/.sage
-    HOME={envdir}
+                             HOME={envdir}
     # We supply pip options by environment variables so that they
     # apply both to the installation of the dependencies and of the package
     sagewheels:              PIP_FIND_LINKS=file://{env:SAGE_SPKG_WHEELS:{env:SAGE_VENV:{toxinidir}/../../../../venv}/var/lib/sage/wheels}
@@ -48,7 +49,16 @@ commands =
     #bash -c 'cd {temp_dir} && SAGE_SRC=$(python -c "from sage.env import SAGE_SRC; print(SAGE_SRC)") && sage-runtests --environment=sage.all__sagemath_objects --initial --optional=sage $SAGE_SRC/sage/structure || echo "(lots of doctest failures are expected)"'
 
 [testenv:sagepython]
-passenv =
-    SAGE_VENV
+basepython = {env:SAGE_VENV}/bin/python3
+
+[testenv:sagepython-sagewheels-nopypi]
+basepython = {env:SAGE_VENV}/bin/python3
+
+[testenv:sagepython-sagewheels-nopypi-norequirements]
+basepython = {env:SAGE_VENV}/bin/python3
+
+[testenv:sagepython-sagewheels]
+basepython = {env:SAGE_VENV}/bin/python3
 
+[testenv:sagepython-norequirements]
 basepython = {env:SAGE_VENV}/bin/python3
diff --git a/pkgs/sagemath-repl/tox.ini b/pkgs/sagemath-repl/tox.ini
index a56aa5bc9fe..b564bcda707 100644
--- a/pkgs/sagemath-repl/tox.ini
+++ b/pkgs/sagemath-repl/tox.ini
@@ -23,14 +23,15 @@ passenv =
                              PKG_CONFIG_PATH
     # Parallel build
                              SAGE_NUM_THREADS
-    # SAGE_VENV only for finding the wheels
-    sagewheels:              SAGE_VENV
+                             SAGE_NUM_THREADS_PARALLEL
+    # SAGE_VENV only for referring to the basepython or finding the wheels
+    sagepython, sagewheels:  SAGE_VENV
     # Location of the wheels
     sagewheels:              SAGE_SPKG_WHEELS
 
 setenv =
     # Sage scripts such as sage-runtests like to use $HOME/.sage
-    HOME={envdir}
+                             HOME={envdir}
     # We supply pip options by environment variables so that they
     # apply both to the installation of the dependencies and of the package
     sagewheels:              PIP_FIND_LINKS=file://{env:SAGE_SPKG_WHEELS:{env:SAGE_VENV:{toxinidir}/../../../../venv}/var/lib/sage/wheels}
@@ -46,7 +47,16 @@ commands =
     bash -c 'cd bin && SAGE_SRC=$({envpython} -c "from sage.env import SAGE_SRC; print(SAGE_SRC)") && sage-runtests --environment=sage.all__sagemath_repl --initial --optional=sage $SAGE_SRC/sage/repl $SAGE_SRC/sage/doctest $SAGE_SRC/sage/misc/sage_input.py $SAGE_SRC/sage/misc/sage_eval.py || echo "(lots of doctest failures are expected)"'
 
 [testenv:sagepython]
-passenv =
-    SAGE_VENV
+basepython = {env:SAGE_VENV}/bin/python3
+
+[testenv:sagepython-sagewheels-nopypi]
+basepython = {env:SAGE_VENV}/bin/python3
+
+[testenv:sagepython-sagewheels-nopypi-norequirements]
+basepython = {env:SAGE_VENV}/bin/python3
+
+[testenv:sagepython-sagewheels]
+basepython = {env:SAGE_VENV}/bin/python3
 
+[testenv:sagepython-norequirements]
 basepython = {env:SAGE_VENV}/bin/python3
diff --git a/pkgs/sagemath-standard/tox.ini b/pkgs/sagemath-standard/tox.ini
index b80af350be4..f021a7d2b22 100644
--- a/pkgs/sagemath-standard/tox.ini
+++ b/pkgs/sagemath-standard/tox.ini
@@ -82,10 +82,9 @@ passenv =
                              PKG_CONFIG_PATH
     # Parallel build
                              SAGE_NUM_THREADS
-    # SAGE_VENV only for finding the wheels
+    # SAGE_VENV only for referring to the basepython or finding the wheels
     sagepython, sagewheels:  SAGE_VENV
-    # Location of the wheels (needs to include a PEP 503 compliant
-    # Simple Repository index, i.e., a subdirectory "simple")
+    # Location of the wheels
     sagewheels:              SAGE_SPKG_WHEELS
 
 setenv =