diff --git a/cirq-core/cirq/ops/dense_pauli_string_test.py b/cirq-core/cirq/ops/dense_pauli_string_test.py
index 99dbed21b8a..44951cab48d 100644
--- a/cirq-core/cirq/ops/dense_pauli_string_test.py
+++ b/cirq-core/cirq/ops/dense_pauli_string_test.py
@@ -376,9 +376,11 @@ def test_protocols():
     cirq.testing.assert_implements_consistent_protocols(-cirq.DensePauliString('Z'))
     cirq.testing.assert_implements_consistent_protocols(1j * cirq.DensePauliString('X'))
     cirq.testing.assert_implements_consistent_protocols(2 * cirq.DensePauliString('X'))
-    cirq.testing.assert_implements_consistent_protocols(t * cirq.DensePauliString('XYIZ'))
     cirq.testing.assert_implements_consistent_protocols(
-        cirq.DensePauliString('XYIZ', coefficient=t + 2)
+        t * cirq.DensePauliString('XYIZ'), ignore_decompose_to_default_gateset=True
+    )
+    cirq.testing.assert_implements_consistent_protocols(
+        cirq.DensePauliString('XYIZ', coefficient=t + 2), ignore_decompose_to_default_gateset=True
     )
     cirq.testing.assert_implements_consistent_protocols(-cirq.DensePauliString('XYIZ'))
     cirq.testing.assert_implements_consistent_protocols(
diff --git a/cirq-core/cirq/ops/random_gate_channel_test.py b/cirq-core/cirq/ops/random_gate_channel_test.py
index ca47f1d023a..6bda121d98d 100644
--- a/cirq-core/cirq/ops/random_gate_channel_test.py
+++ b/cirq-core/cirq/ops/random_gate_channel_test.py
@@ -80,16 +80,20 @@ def test_eq():
 
 def test_consistent_protocols():
     cirq.testing.assert_implements_consistent_protocols(
-        cirq.RandomGateChannel(sub_gate=cirq.X, probability=1)
+        cirq.RandomGateChannel(sub_gate=cirq.X, probability=1),
+        ignore_decompose_to_default_gateset=True,
     )
     cirq.testing.assert_implements_consistent_protocols(
-        cirq.RandomGateChannel(sub_gate=cirq.X, probability=0)
+        cirq.RandomGateChannel(sub_gate=cirq.X, probability=0),
+        ignore_decompose_to_default_gateset=True,
     )
     cirq.testing.assert_implements_consistent_protocols(
-        cirq.RandomGateChannel(sub_gate=cirq.X, probability=sympy.Symbol('x') / 2)
+        cirq.RandomGateChannel(sub_gate=cirq.X, probability=sympy.Symbol('x') / 2),
+        ignore_decompose_to_default_gateset=True,
     )
     cirq.testing.assert_implements_consistent_protocols(
-        cirq.RandomGateChannel(sub_gate=cirq.X, probability=0.5)
+        cirq.RandomGateChannel(sub_gate=cirq.X, probability=0.5),
+        ignore_decompose_to_default_gateset=True,
     )
 
 
diff --git a/cirq-core/cirq/testing/consistent_decomposition.py b/cirq-core/cirq/testing/consistent_decomposition.py
index c66eead8c2f..4ded1ec9525 100644
--- a/cirq-core/cirq/testing/consistent_decomposition.py
+++ b/cirq-core/cirq/testing/consistent_decomposition.py
@@ -53,6 +53,8 @@ def _known_gate_with_no_decomposition(val: Any):
     """Checks whether `val` is a known gate with no default decomposition to default gateset."""
     if isinstance(val, ops.MatrixGate):
         return protocols.qid_shape(val) not in [(2,), (2,) * 2, (2,) * 3]
+    if isinstance(val, ops.BaseDensePauliString) and not protocols.has_unitary(val):
+        return True
     if isinstance(val, ops.ControlledGate):
         if protocols.is_parameterized(val):
             return True
diff --git a/cirq-core/cirq/testing/consistent_protocols.py b/cirq-core/cirq/testing/consistent_protocols.py
index 51d44a77416..8a759384b58 100644
--- a/cirq-core/cirq/testing/consistent_protocols.py
+++ b/cirq-core/cirq/testing/consistent_protocols.py
@@ -26,6 +26,7 @@
 )
 from cirq.testing.consistent_decomposition import (
     assert_decompose_is_consistent_with_unitary,
+    assert_decompose_ends_at_default_gateset,
 )
 from cirq.testing.consistent_phase_by import (
     assert_phase_by_is_consistent_with_unitary,
@@ -55,6 +56,7 @@ def assert_implements_consistent_protocols(
     setup_code: str = 'import cirq\nimport numpy as np\nimport sympy',
     global_vals: Optional[Dict[str, Any]] = None,
     local_vals: Optional[Dict[str, Any]] = None,
+    ignore_decompose_to_default_gateset: bool = False,
 ) -> None:
     """Checks that a value is internally consistent and has a good __repr__."""
     global_vals = global_vals or {}
@@ -66,6 +68,7 @@ def assert_implements_consistent_protocols(
         setup_code=setup_code,
         global_vals=global_vals,
         local_vals=local_vals,
+        ignore_decompose_to_default_gateset=ignore_decompose_to_default_gateset,
     )
 
     for exponent in exponents:
@@ -77,6 +80,7 @@ def assert_implements_consistent_protocols(
                 setup_code=setup_code,
                 global_vals=global_vals,
                 local_vals=local_vals,
+                ignore_decompose_to_default_gateset=ignore_decompose_to_default_gateset,
             )
 
 
@@ -90,6 +94,7 @@ def assert_eigengate_implements_consistent_protocols(
     setup_code: str = 'import cirq\nimport numpy as np\nimport sympy',
     global_vals: Optional[Dict[str, Any]] = None,
     local_vals: Optional[Dict[str, Any]] = None,
+    ignore_decompose_to_default_gateset: bool = False,
 ) -> None:
     """Checks that an EigenGate subclass is internally consistent and has a
     good __repr__."""
@@ -105,6 +110,7 @@ def assert_eigengate_implements_consistent_protocols(
                 setup_code=setup_code,
                 global_vals=global_vals,
                 local_vals=local_vals,
+                ignore_decompose_to_default_gateset=ignore_decompose_to_default_gateset,
             )
 
 
@@ -143,6 +149,7 @@ def _assert_meets_standards_helper(
     setup_code: str,
     global_vals: Optional[Dict[str, Any]],
     local_vals: Optional[Dict[str, Any]],
+    ignore_decompose_to_default_gateset: bool,
 ) -> None:
     __tracebackhide__ = True  # pylint: disable=unused-variable
 
@@ -154,6 +161,8 @@ def _assert_meets_standards_helper(
     assert_qasm_is_consistent_with_unitary(val)
     assert_has_consistent_trace_distance_bound(val)
     assert_decompose_is_consistent_with_unitary(val, ignoring_global_phase=ignoring_global_phase)
+    if not ignore_decompose_to_default_gateset:
+        assert_decompose_ends_at_default_gateset(val)
     assert_phase_by_is_consistent_with_unitary(val)
     assert_pauli_expansion_is_consistent_with_unitary(val)
     assert_equivalent_repr(
diff --git a/cirq-core/cirq/testing/consistent_protocols_test.py b/cirq-core/cirq/testing/consistent_protocols_test.py
index a8ef6e8f655..8543f3ae16c 100644
--- a/cirq-core/cirq/testing/consistent_protocols_test.py
+++ b/cirq-core/cirq/testing/consistent_protocols_test.py
@@ -65,9 +65,6 @@ def _decompose_(self, qubits: Sequence[cirq.Qid]) -> cirq.OP_TREE:
         q = qubits[0]
         z = cirq.Z(q) ** self.phase_exponent
         x = cirq.X(q) ** self.exponent
-        if cirq.is_parameterized(z):
-            # coverage: ignore
-            return NotImplemented
         return z ** -1, x, z
 
     def _pauli_expansion_(self) -> cirq.LinearDict[str]:
@@ -260,12 +257,16 @@ def test_assert_implements_consistent_protocols():
 
 def test_assert_eigengate_implements_consistent_protocols():
     cirq.testing.assert_eigengate_implements_consistent_protocols(
-        GoodEigenGate, global_vals={'GoodEigenGate': GoodEigenGate}
+        GoodEigenGate,
+        global_vals={'GoodEigenGate': GoodEigenGate},
+        ignore_decompose_to_default_gateset=True,
     )
 
     with pytest.raises(AssertionError):
         cirq.testing.assert_eigengate_implements_consistent_protocols(
-            BadEigenGate, global_vals={'BadEigenGate': BadEigenGate}
+            BadEigenGate,
+            global_vals={'BadEigenGate': BadEigenGate},
+            ignore_decompose_to_default_gateset=True,
         )
 
 
diff --git a/cirq-google/cirq_google/experimental/ops/coupler_pulse_test.py b/cirq-google/cirq_google/experimental/ops/coupler_pulse_test.py
index b91973120dc..8878c6ae90e 100644
--- a/cirq-google/cirq_google/experimental/ops/coupler_pulse_test.py
+++ b/cirq-google/cirq_google/experimental/ops/coupler_pulse_test.py
@@ -25,6 +25,7 @@ def test_consistent_protocols():
         gate,
         setup_code='import cirq\nimport numpy as np\nimport sympy\nimport cirq_google',
         qubit_count=2,
+        ignore_decompose_to_default_gateset=True,
     )
     assert gate.num_qubits() == 2