REF: implement _should_compare

pandas/core/indexes/base.py

@@ -4938,6 +4938,26 @@ def get_indexer_for(self, target, **kwargs):
         indexer, _ = self.get_indexer_non_unique(target)
         return indexer
 
+    def _get_indexer_non_comparable(self, target, method, unique: bool = True):
+        """
+        If we have non-comparable dtypes, there will never be any matches.
+
+        For get_indexer calls with a method, this is an _inequality_ check
+        which raises TypeError.
+        """
+        if method is not None:
+            other = self._get_other_deep(target)
+            raise TypeError(f"Cannot compare dtypes {self.dtype} and {other.dtype}")
+
+        no_matches = -1 * np.ones(target.shape, dtype=np.intp)
+        if unique:
+            # This is for get_indexer
+            return no_matches
+        else:
+            # This is for get_indexer_non_unique
+            missing = np.arange(len(target), dtype=np.intp)
+            return no_matches, missing
+
     @property
     def _index_as_unique(self):
         """
@@ -4973,6 +4993,22 @@ def _maybe_promote(self, other: "Index"):
 
         return self, other
 
+    def _get_other_deep(self, other: "Index") -> "Index":
+        dtype = other.dtype
+        if is_categorical_dtype(dtype):
+            # If there is ever a SparseIndex, this could get dispatched
+            #  here too.
+            return dtype.categories
+        return other
+
+    def _should_compare(self, other: "Index") -> bool:
+        """
+        Check if `self == other` can ever have non-False entries.
+        """
+        other = self._get_other_deep(other)
+        dtype = other.dtype
+        return self._is_comparable_dtype(dtype) or is_object_dtype(dtype)
+
     def _is_comparable_dtype(self, dtype: DtypeObj) -> bool:
         """
         Can we compare values of the given dtype to our own?

pandas/core/indexes/period.py

@@ -452,13 +452,10 @@ def join(self, other, how="left", level=None, return_indexers=False, sort=False)
     def get_indexer(self, target, method=None, limit=None, tolerance=None):
         target = ensure_index(target)
 
-        if isinstance(target, PeriodIndex):
-            if not self._is_comparable_dtype(target.dtype):
-                # i.e. target.freq != self.freq
-                # No matches
-                no_matches = -1 * np.ones(self.shape, dtype=np.intp)
-                return no_matches
+        if not self._should_compare(target):
+            return self._get_indexer_non_comparable(target, method, unique=True)
 
+        if isinstance(target, PeriodIndex):
             target = target._get_engine_target()  # i.e. target.asi8
             self_index = self._int64index
         else:

pandas/tests/indexes/period/test_indexing.py

@@ -21,6 +21,28 @@
 )
 import pandas._testing as tm
 
+dti4 = date_range("2016-01-01", periods=4)
+dti = dti4[:-1]
+rng = pd.Index(range(3))
+
+
+@pytest.fixture(
+    params=[
+        dti,
+        dti.tz_localize("UTC"),
+        dti.to_period("W"),
+        dti - dti[0],
+        rng,
+        pd.Index([1, 2, 3]),
+        pd.Index([2.0, 3.0, 4.0]),
+        pd.Index([4, 5, 6], dtype="u8"),
+        pd.IntervalIndex.from_breaks(dti4),
+    ]
+)
+def non_comparable(request):
+    # All have length 3
+    return request.param
+
 
 class TestGetItem:
     def test_ellipsis(self):
@@ -438,6 +460,37 @@ def test_get_indexer_mismatched_dtype(self):
         result = pi.get_indexer_non_unique(pi2)[0]
         tm.assert_numpy_array_equal(result, expected)
 
+    def test_get_indexer_mismatched_dtype_different_length(self, non_comparable):
+        # without method we arent checking inequalities, so get all-missing
+        #  but do not raise
+        dti = date_range("2016-01-01", periods=3)
+        pi = dti.to_period("D")
+
+        other = non_comparable
+
+        res = pi[:-1].get_indexer(other)
+        expected = -np.ones(other.shape, dtype=np.intp)
+        tm.assert_numpy_array_equal(res, expected)
+
+    @pytest.mark.parametrize("method", ["pad", "backfill", "nearest"])
+    def test_get_indexer_mismatched_dtype_with_method(self, non_comparable, method):
+        dti = date_range("2016-01-01", periods=3)
+        pi = dti.to_period("D")
+
+        other = non_comparable
+
+        msg = re.escape(f"Cannot compare dtypes {pi.dtype} and {other.dtype}")
+        with pytest.raises(TypeError, match=msg):
+            pi.get_indexer(other, method=method)
+
+        for dtype in ["object", "category"]:
+            other2 = other.astype(dtype)
+            if dtype == "object" and isinstance(other, PeriodIndex):
+                continue
+            # For object dtype we are liable to get a different exception message
+            with pytest.raises(TypeError):
+                pi.get_indexer(other2, method=method)
+
     def test_get_indexer_non_unique(self):
         # GH 17717
         p1 = Period("2017-09-02")