diff --git a/modin/core/dataframe/algebra/binary.py b/modin/core/dataframe/algebra/binary.py
index b5e701d2d4b..b107089eda5 100644
--- a/modin/core/dataframe/algebra/binary.py
+++ b/modin/core/dataframe/algebra/binary.py
@@ -298,6 +298,7 @@ def register(
         cls,
         func: Callable[..., pandas.DataFrame],
         join_type: str = "outer",
+        sort: bool = None,
         labels: str = "replace",
         infer_dtypes: Optional[str] = None,
     ) -> Callable[..., PandasQueryCompiler]:
@@ -310,6 +311,8 @@ def register(
             Binary function to execute. Have to be able to accept at least two arguments.
         join_type : {'left', 'right', 'outer', 'inner', None}, default: 'outer'
             Type of join that will be used if indices of operands are not aligned.
+        sort : bool, default: None
+            Whether to sort index and columns or not.
         labels : {"keep", "replace", "drop"}, default: "replace"
             Whether keep labels from left Modin DataFrame, replace them with labels
             from joined DataFrame or drop altogether to make them be computed lazily later.
@@ -419,6 +422,7 @@ def caller(
                             lambda x, y: func(x, y, *args, **kwargs),
                             [other._modin_frame],
                             join_type=join_type,
+                            sort=sort,
                             labels=labels,
                             dtypes=dtypes,
                         ),
diff --git a/modin/core/dataframe/pandas/dataframe/dataframe.py b/modin/core/dataframe/pandas/dataframe/dataframe.py
index b8467ebe084..5456f28f127 100644
--- a/modin/core/dataframe/pandas/dataframe/dataframe.py
+++ b/modin/core/dataframe/pandas/dataframe/dataframe.py
@@ -3255,7 +3255,6 @@ def broadcast_apply(
                 axis,
                 other,
                 join_type,
-                sort=not self.get_axis(axis).equals(other.get_axis(axis)),
             )
             # unwrap list returned by `copartition`.
             right_parts = right_parts[0]
@@ -3681,7 +3680,7 @@ def _check_if_axes_identical(self, other: PandasDataframe, axis: int = 0) -> boo
         ) and self._get_axis_lengths(axis) == other._get_axis_lengths(axis)
 
     def _copartition(
-        self, axis, other, how, sort, force_repartition=False, fill_value=None
+        self, axis, other, how, sort=None, force_repartition=False, fill_value=None
     ):
         """
         Copartition two Modin DataFrames.
@@ -3696,8 +3695,9 @@ def _copartition(
             Other Modin DataFrame(s) to copartition against.
         how : str
             How to manage joining the index object ("left", "right", etc.).
-        sort : bool
+        sort : bool, default: None
             Whether sort the joined index or not.
+            If ``None``, sort is defined in depend on labels equality along the axis.
         force_repartition : bool, default: False
             Whether force the repartitioning or not. By default,
             this method will skip repartitioning if it is possible. This is because
@@ -3730,6 +3730,9 @@ def _copartition(
                 self._get_axis_lengths_cache(axis),
             )
 
+        if sort is None:
+            sort = not all(self.get_axis(axis).equals(o.get_axis(axis)) for o in other)
+
         self_index = self.get_axis(axis)
         others_index = [o.get_axis(axis) for o in other]
         joined_index, make_reindexer = self._join_index_objects(
@@ -3823,6 +3826,7 @@ def n_ary_op(
         op,
         right_frames: list[PandasDataframe],
         join_type="outer",
+        sort=None,
         copartition_along_columns=True,
         labels="replace",
         dtypes: Optional[pandas.Series] = None,
@@ -3838,6 +3842,8 @@ def n_ary_op(
             Modin DataFrames to join with.
         join_type : str, default: "outer"
             Type of join to apply.
+        sort : bool, default: None
+            Whether to sort index and columns or not.
         copartition_along_columns : bool, default: True
             Whether to perform copartitioning along columns or not.
             For some ops this isn't needed (e.g., `fillna`).
@@ -3854,7 +3860,10 @@ def n_ary_op(
             New Modin DataFrame.
         """
         left_parts, list_of_right_parts, joined_index, row_lengths = self._copartition(
-            0, right_frames, join_type, sort=True
+            0,
+            right_frames,
+            join_type,
+            sort=sort,
         )
         if copartition_along_columns:
             new_left_frame = self.__constructor__(
@@ -3886,7 +3895,7 @@ def n_ary_op(
                 1,
                 new_right_frames,
                 join_type,
-                sort=True,
+                sort=sort,
             )
         else:
             joined_columns = self.copy_columns_cache(copy_lengths=True)
@@ -3978,7 +3987,7 @@ def _compute_new_widths():
                 joined_index,
                 partition_sizes_along_axis,
             ) = self._copartition(
-                axis.value ^ 1, others, how, sort, force_repartition=False
+                axis.value ^ 1, others, how, sort=sort, force_repartition=False
             )
             if axis == Axis.COL_WISE:
                 new_lengths = partition_sizes_along_axis
diff --git a/modin/core/storage_formats/pandas/query_compiler.py b/modin/core/storage_formats/pandas/query_compiler.py
index 7c4f7e79f55..9d4467c2085 100644
--- a/modin/core/storage_formats/pandas/query_compiler.py
+++ b/modin/core/storage_formats/pandas/query_compiler.py
@@ -460,6 +460,7 @@ def to_numpy(self, **kwargs):
     df_update = Binary.register(
         copy_df_for_func(pandas.DataFrame.update, display_name="update"),
         join_type="left",
+        sort=False,
     )
     series_update = Binary.register(
         copy_df_for_func(
@@ -467,6 +468,7 @@ def to_numpy(self, **kwargs):
             display_name="update",
         ),
         join_type="left",
+        sort=False,
     )
 
     # Needed for numpy API
diff --git a/modin/tests/pandas/dataframe/test_binary.py b/modin/tests/pandas/dataframe/test_binary.py
index e153f9f892f..108e2620aac 100644
--- a/modin/tests/pandas/dataframe/test_binary.py
+++ b/modin/tests/pandas/dataframe/test_binary.py
@@ -527,3 +527,30 @@ def test_arithmetic_with_tricky_dtypes(val1, val2, op, request):
         lambda dfs: getattr(dfs[0], op)(dfs[1]),
         expected_exception=expected_exception,
     )
+
+
+@pytest.mark.parametrize(
+    "data, other_data",
+    [
+        ({"A": [1, 2, 3], "B": [400, 500, 600]}, {"B": [4, 5, 6], "C": [7, 8, 9]}),
+        ({"C": [1, 2, 3], "B": [400, 500, 600]}, {"B": [4, 5, 6], "A": [7, 8, 9]}),
+    ],
+)
+@pytest.mark.parametrize("axis", [0, 1])
+@pytest.mark.parametrize("match_index", [True, False])
+def test_bin_op_mismatched_columns(data, other_data, axis, match_index):
+    modin_df, pandas_df = create_test_dfs(data)
+    other_modin_df, other_pandas_df = create_test_dfs(other_data)
+    if axis == 0:
+        if not match_index:
+            modin_df.index = pandas_df.index = ["1", "2", "3"]
+            other_modin_df.index = other_pandas_df.index = ["2", "1", "3"]
+    eval_general(
+        modin_df,
+        pandas_df,
+        lambda df: (
+            df.add(other_modin_df, axis=axis)
+            if isinstance(df, pd.DataFrame)
+            else df.add(other_pandas_df, axis=axis)
+        ),
+    )
diff --git a/modin/tests/pandas/dataframe/test_map_metadata.py b/modin/tests/pandas/dataframe/test_map_metadata.py
index 4b19d5fbd9d..d6980cd6761 100644
--- a/modin/tests/pandas/dataframe/test_map_metadata.py
+++ b/modin/tests/pandas/dataframe/test_map_metadata.py
@@ -1592,6 +1592,7 @@ def test_transpose(data):
     "data, other_data",
     [
         ({"A": [1, 2, 3], "B": [400, 500, 600]}, {"B": [4, 5, 6], "C": [7, 8, 9]}),
+        ({"C": [1, 2, 3], "B": [400, 500, 600]}, {"B": [4, 5, 6], "A": [7, 8, 9]}),
         (
             {"A": ["a", "b", "c"], "B": ["x", "y", "z"]},
             {"B": ["d", "e", "f", "g", "h", "i"]},