BUG: String[pyarrow] comparison with mixed object

pandas/core/arrays/arrow/array.py

@@ -829,12 +829,17 @@ def _cmp_method(self, other, op) -> ArrowExtensionArray:
         pc_func = ARROW_CMP_FUNCS[op.__name__]
         if isinstance(other, (ExtensionArray, np.ndarray, list)):
             try:
-                result = pc_func(self._pa_array, self._box_pa(other))
-            except pa.ArrowNotImplementedError:
-                # TODO: could this be wrong if other is object dtype?
-                #  in which case we need to operate pointwise?
-                result = ops.invalid_comparison(self, other, op)
-                result = pa.array(result, type=pa.bool_())
+                boxed = self._box_pa(other)
+            except pa.lib.ArrowInvalid:
+                # e.g. GH#60228 [1, "b"] we have to operate pointwise
+                res_values = [op(x, y) for x, y in zip(self, other)]
+                result = pa.array(res_values, type=pa.bool_(), from_pandas=True)
+            else:
+                try:
+                    result = pc_func(self._pa_array, boxed)
+                except pa.ArrowNotImplementedError:
+                    result = ops.invalid_comparison(self, other, op)
+                    result = pa.array(result, type=pa.bool_())
         elif is_scalar(other):
             try:
                 result = pc_func(self._pa_array, self._box_pa(other))

pandas/tests/extension/test_string.py

@@ -288,3 +288,19 @@ def test_searchsorted_with_na_raises(data_for_sorting, as_series):
     )
     with pytest.raises(ValueError, match=msg):
         arr.searchsorted(b)
+
+
+def test_mixed_object_comparison(dtype):
+    # GH#60228
+    ser = pd.Series(["a", "b"], dtype=dtype)
+
+    mixed = pd.Series([1, "b"], dtype=object)
+
+    result = ser == mixed
+    expected = pd.Series([False, True], dtype=bool)
+    if dtype.storage == "python" and dtype.na_value is pd.NA:
+        expected = expected.astype("boolean")
+    elif dtype.storage == "pyarrow" and dtype.na_value is pd.NA:
+        expected = expected.astype("bool[pyarrow]")
+
+    tm.assert_series_equal(result, expected)