diff --git a/xarray/core/variable.py b/xarray/core/variable.py
index a96adb31e64..52125ec4113 100644
--- a/xarray/core/variable.py
+++ b/xarray/core/variable.py
@@ -2813,6 +2813,11 @@ def name(self):
     def name(self, value):
         raise AttributeError("cannot modify name of IndexVariable in-place")
 
+    def _inplace_binary_op(self, other, f):
+        raise TypeError(
+            "Values of an IndexVariable are immutable and can not be modified inplace"
+        )
+
 
 # for backwards compatibility
 Coordinate = utils.alias(IndexVariable, "Coordinate")
diff --git a/xarray/tests/test_dataarray.py b/xarray/tests/test_dataarray.py
index 53c650046e7..5e9c1b87ce2 100644
--- a/xarray/tests/test_dataarray.py
+++ b/xarray/tests/test_dataarray.py
@@ -1987,6 +1987,18 @@ def test_inplace_math_basics(self):
         assert_array_equal(b.values, x)
         assert source_ndarray(b.values) is x
 
+    def test_inplace_math_error(self):
+        data = np.random.rand(4)
+        times = np.arange(4)
+        foo = DataArray(data, coords=[times], dims=["time"])
+        b = times.copy()
+        with pytest.raises(
+            TypeError, match=r"Values of an IndexVariable are immutable"
+        ):
+            foo.coords["time"] += 1
+        # Check error throwing prevented inplace operation
+        assert_array_equal(foo.coords["time"], b)
+
     def test_inplace_math_automatic_alignment(self):
         a = DataArray(range(5), [("x", range(5))])
         b = DataArray(range(1, 6), [("x", range(1, 6))])
diff --git a/xarray/tests/test_variable.py b/xarray/tests/test_variable.py
index 9c0e45c5da9..3267af8b45b 100644
--- a/xarray/tests/test_variable.py
+++ b/xarray/tests/test_variable.py
@@ -1656,6 +1656,14 @@ def test_inplace_math(self):
         with pytest.raises(ValueError, match=r"dimensions cannot change"):
             v += Variable("y", np.arange(5))
 
+    def test_inplace_math_error(self):
+        x = np.arange(5)
+        v = IndexVariable(["x"], x)
+        with pytest.raises(
+            TypeError, match=r"Values of an IndexVariable are immutable"
+        ):
+            v += 1
+
     def test_reduce(self):
         v = Variable(["x", "y"], self.d, {"ignored": "attributes"})
         assert_identical(v.reduce(np.std, "x"), Variable(["y"], self.d.std(axis=0)))