diff --git a/doc/source/whatsnew/v1.0.0.rst b/doc/source/whatsnew/v1.0.0.rst
index a3d75d69e1e82..a78bc07ac2715 100644
--- a/doc/source/whatsnew/v1.0.0.rst
+++ b/doc/source/whatsnew/v1.0.0.rst
@@ -199,7 +199,7 @@ Timezones
 Numeric
 ^^^^^^^
 - Bug in :meth:`DataFrame.quantile` with zero-column :class:`DataFrame` incorrectly raising (:issue:`23925`)
--
+- :class:`DataFrame` inequality comparisons with object-dtype and ``complex`` entries failing to raise ``TypeError`` like their :class:`Series` counterparts (:issue:`28079`)
 -
 
 Conversion
diff --git a/pandas/core/ops/__init__.py b/pandas/core/ops/__init__.py
index 4f027843fbac1..eb901630b753a 100644
--- a/pandas/core/ops/__init__.py
+++ b/pandas/core/ops/__init__.py
@@ -28,7 +28,7 @@
     ABCIndexClass,
     ABCSeries,
 )
-from pandas.core.dtypes.missing import isna, notna
+from pandas.core.dtypes.missing import isna
 
 from pandas._typing import ArrayLike
 from pandas.core.construction import array, extract_array
@@ -354,38 +354,6 @@ def fill_binop(left, right, fill_value):
     return left, right
 
 
-def mask_cmp_op(x, y, op):
-    """
-    Apply the function `op` to only non-null points in x and y.
-
-    Parameters
-    ----------
-    x : array-like
-    y : array-like
-    op : binary operation
-
-    Returns
-    -------
-    result : ndarray[bool]
-    """
-    xrav = x.ravel()
-    result = np.empty(x.size, dtype=bool)
-    if isinstance(y, (np.ndarray, ABCSeries)):
-        yrav = y.ravel()
-        mask = notna(xrav) & notna(yrav)
-        result[mask] = op(np.array(list(xrav[mask])), np.array(list(yrav[mask])))
-    else:
-        mask = notna(xrav)
-        result[mask] = op(np.array(list(xrav[mask])), y)
-
-    if op == operator.ne:  # pragma: no cover
-        np.putmask(result, ~mask, True)
-    else:
-        np.putmask(result, ~mask, False)
-    result = result.reshape(x.shape)
-    return result
-
-
 # -----------------------------------------------------------------------------
 # Dispatch logic
 
@@ -905,14 +873,6 @@ def _flex_comp_method_FRAME(cls, op, special):
     op_name = _get_op_name(op, special)
     default_axis = _get_frame_op_default_axis(op_name)
 
-    def na_op(x, y):
-        try:
-            with np.errstate(invalid="ignore"):
-                result = op(x, y)
-        except TypeError:
-            result = mask_cmp_op(x, y, op)
-        return result
-
     doc = _flex_comp_doc_FRAME.format(
         op_name=op_name, desc=_op_descriptions[op_name]["desc"]
     )
@@ -926,16 +886,16 @@ def f(self, other, axis=default_axis, level=None):
             # Another DataFrame
             if not self._indexed_same(other):
                 self, other = self.align(other, "outer", level=level, copy=False)
-            new_data = dispatch_to_series(self, other, na_op, str_rep)
+            new_data = dispatch_to_series(self, other, op, str_rep)
             return self._construct_result(new_data)
 
         elif isinstance(other, ABCSeries):
             return _combine_series_frame(
-                self, other, na_op, fill_value=None, axis=axis, level=level
+                self, other, op, fill_value=None, axis=axis, level=level
             )
         else:
             # in this case we always have `np.ndim(other) == 0`
-            new_data = dispatch_to_series(self, other, na_op)
+            new_data = dispatch_to_series(self, other, op)
             return self._construct_result(new_data)
 
     f.__name__ = op_name
diff --git a/pandas/tests/frame/test_arithmetic.py b/pandas/tests/frame/test_arithmetic.py
index fc3640503e385..3b46e834933b3 100644
--- a/pandas/tests/frame/test_arithmetic.py
+++ b/pandas/tests/frame/test_arithmetic.py
@@ -235,21 +235,46 @@ def _test_seq(df, idx_ser, col_ser):
         rs = df.le(df)
         assert not rs.loc[0, 0]
 
+    def test_bool_flex_frame_complex_dtype(self):
         # complex
         arr = np.array([np.nan, 1, 6, np.nan])
         arr2 = np.array([2j, np.nan, 7, None])
         df = pd.DataFrame({"a": arr})
         df2 = pd.DataFrame({"a": arr2})
-        rs = df.gt(df2)
-        assert not rs.values.any()
+
+        msg = "|".join(
+            [
+                "'>' not supported between instances of '.*' and 'complex'",
+                r"unorderable types: .*complex\(\)",  # PY35
+            ]
+        )
+        with pytest.raises(TypeError, match=msg):
+            # inequalities are not well-defined for complex numbers
+            df.gt(df2)
+        with pytest.raises(TypeError, match=msg):
+            # regression test that we get the same behavior for Series
+            df["a"].gt(df2["a"])
+        with pytest.raises(TypeError, match=msg):
+            # Check that we match numpy behavior here
+            df.values > df2.values
+
         rs = df.ne(df2)
         assert rs.values.all()
 
         arr3 = np.array([2j, np.nan, None])
         df3 = pd.DataFrame({"a": arr3})
-        rs = df3.gt(2j)
-        assert not rs.values.any()
 
+        with pytest.raises(TypeError, match=msg):
+            # inequalities are not well-defined for complex numbers
+            df3.gt(2j)
+        with pytest.raises(TypeError, match=msg):
+            # regression test that we get the same behavior for Series
+            df3["a"].gt(2j)
+        with pytest.raises(TypeError, match=msg):
+            # Check that we match numpy behavior here
+            df3.values > 2j
+
+    def test_bool_flex_frame_object_dtype(self):
         # corner, dtype=object
         df1 = pd.DataFrame({"col": ["foo", np.nan, "bar"]})
         df2 = pd.DataFrame({"col": ["foo", datetime.now(), "bar"]})