BUG: __eq__ raising for new arrow string dtype for incompatible objects

doc/source/whatsnew/v2.2.0.rst

@@ -502,6 +502,7 @@ Strings
 - Bug in :func:`pandas.api.types.is_string_dtype` while checking object array with no elements is of the string dtype (:issue:`54661`)
 - Bug in :meth:`DataFrame.apply` failing when ``engine="numba"`` and columns or index have ``StringDtype`` (:issue:`56189`)
 - Bug in :meth:`Series.str.startswith` and :meth:`Series.str.endswith` with arguments of type ``tuple[str, ...]`` for ``string[pyarrow]`` (:issue:`54942`)
+- Bug in comparison operations for ``dtype="string[pyarrow_numpy]"`` raising if dtypes can't be compared (:issue:`56008`)
 
 Interval
 ^^^^^^^^

pandas/core/arrays/string_arrow.py

@@ -40,6 +40,7 @@
     BaseStringArray,
     StringDtype,
 )
+from pandas.core.ops import invalid_comparison
 from pandas.core.strings.object_array import ObjectStringArrayMixin
 
 if not pa_version_under10p1:
@@ -662,7 +663,10 @@ def _convert_int_dtype(self, result):
         return result
 
     def _cmp_method(self, other, op):
-        result = super()._cmp_method(other, op)
+        try:
+            result = super()._cmp_method(other, op)
+        except pa.lib.ArrowNotImplementedError:
+            return invalid_comparison(self, other, op)
         return result.to_numpy(np.bool_, na_value=False)
 
     def value_counts(self, dropna: bool = True) -> Series:

pandas/tests/series/test_logical_ops.py

@@ -530,3 +530,18 @@ def test_int_dtype_different_index_not_bool(self):
 
         result = ser1 ^ ser2
         tm.assert_series_equal(result, expected)
+
+    def test_pyarrow_numpy_string_invalid(self):
+        # GH#56008
+        ser = Series([False, True])
+        ser2 = Series(["a", "b"], dtype="string[pyarrow_numpy]")
+        result = ser == ser2
+        expected = Series(False, index=ser.index)
+        tm.assert_series_equal(result, expected)
+
+        result = ser != ser2
+        expected = Series(True, index=ser.index)
+        tm.assert_series_equal(result, expected)
+
+        with pytest.raises(TypeError, match="Invalid comparison"):
+            ser > ser2