diff --git a/xarray/core/datatree.py b/xarray/core/datatree.py
index bd583ac86cb..27a2da88701 100644
--- a/xarray/core/datatree.py
+++ b/xarray/core/datatree.py
@@ -374,7 +374,7 @@ def map(  # type: ignore[override]
 
 
 class DataTree(
-    NamedNode,
+    NamedNode["DataTree"],
     MappedDatasetMethodsMixin,
     MappedDataWithCoords,
     DataTreeArithmeticMixin,
@@ -486,6 +486,35 @@ def _pre_attach(self: DataTree, parent: DataTree, name: str) -> None:
         parent_ds = parent._to_dataset_view(rebuild_dims=False, inherited=True)
         check_alignment(path, node_ds, parent_ds, self.children)
 
+    def _dedup_inherited_coordinates(self):
+        # This method removes repeated indexes (and correpsonding coordinates)
+        # that are repeated between a DataTree and its parents.
+        #
+        # TODO(shoyer): Decide how to handle repeated coordinates *without* an
+        # index. Should these be allowed, in which case we probably want to
+        # exclude them from inheritance, or should they be automatically
+        # dropped?
+        # https://github.com/pydata/xarray/issues/9475#issuecomment-2357004264
+        removed_something = False
+        for name in self._indexes.parents:
+            if name in self._node_indexes:
+                # Indexes on a Dataset always have a corresponding coordinate
+                del self._node_indexes[name]
+                del self._node_coord_variables[name]
+                removed_something = True
+
+        if removed_something:
+            self._node_dims = calculate_dimensions(
+                self._data_variables | self._node_coord_variables
+            )
+
+        for child in self._children.values():
+            child._dedup_inherited_coordinates()
+
+    def _post_attach(self: DataTree, parent: DataTree, name: str) -> None:
+        super()._post_attach(parent, name)
+        self._dedup_inherited_coordinates()
+
     @property
     def _coord_variables(self) -> ChainMap[Hashable, Variable]:
         return ChainMap(
diff --git a/xarray/tests/test_datatree.py b/xarray/tests/test_datatree.py
index 4e22ba57e2e..e009e7c47ab 100644
--- a/xarray/tests/test_datatree.py
+++ b/xarray/tests/test_datatree.py
@@ -1190,6 +1190,22 @@ def test_inherited_coords_override(self):
         xr.testing.assert_equal(dt["/b/y"], xr.DataArray(2, coords=sub_coords))
         xr.testing.assert_equal(dt["/b/z"], xr.DataArray(3, coords=sub_coords))
 
+    def test_inherited_coords_with_index_are_deduplicated(self):
+        dt = DataTree.from_dict(
+            {
+                "/": xr.Dataset(coords={"x": [1, 2]}),
+                "/b": xr.Dataset(coords={"x": [1, 2]}),
+            }
+        )
+        child_dataset = dt.children["b"].to_dataset(inherited=False)
+        expected = xr.Dataset()
+        assert_identical(child_dataset, expected)
+
+        dt["/c"] = xr.Dataset({"foo": ("x", [4, 5])}, coords={"x": [1, 2]})
+        child_dataset = dt.children["c"].to_dataset(inherited=False)
+        expected = xr.Dataset({"foo": ("x", [4, 5])})
+        assert_identical(child_dataset, expected)
+
     def test_inconsistent_dims(self):
         expected_msg = _exact_match(
             """
@@ -1534,6 +1550,18 @@ def test_binary_op_on_datatree(self):
         result = dt * dt  # type: ignore[operator]
         assert_equal(result, expected)
 
+    def test_arithmetic_inherited_coords(self):
+        tree = DataTree(xr.Dataset(coords={"x": [1, 2, 3]}))
+        tree["/foo"] = DataTree(xr.Dataset({"bar": ("x", [4, 5, 6])}))
+        actual = 2 * tree  # type: ignore
+
+        actual_dataset = actual.children["foo"].to_dataset(inherited=False)
+        assert "x" not in actual_dataset.coords
+
+        expected = tree.copy()
+        expected["/foo/bar"].data = np.array([8, 10, 12])
+        assert_identical(actual, expected)
+
 
 class TestUFuncs:
     def test_tree(self, create_test_datatree):