diff --git a/lib/iris/cube.py b/lib/iris/cube.py
index a1a07caab2..bf23233f4c 100644
--- a/lib/iris/cube.py
+++ b/lib/iris/cube.py
@@ -1095,7 +1095,7 @@ def add_cell_measure(self, cell_measure, data_dims=None):
         if self.cell_measures(cell_measure):
             raise ValueError("Duplicate cell_measures are not permitted.")
         data_dims = self._check_multi_dim_metadata(cell_measure, data_dims)
-        self._cell_measures_and_dims.append([cell_measure, data_dims])
+        self._cell_measures_and_dims.append((cell_measure, data_dims))
         self._cell_measures_and_dims.sort(
             key=lambda cm_dims: (cm_dims[0]._as_defn(), cm_dims[1])
         )
@@ -1125,7 +1125,7 @@ def add_ancillary_variable(self, ancillary_variable, data_dims=None):
             ancillary_variable, data_dims
         )
         self._ancillary_variables_and_dims.append(
-            [ancillary_variable, data_dims]
+            (ancillary_variable, data_dims)
         )
         self._ancillary_variables_and_dims.sort(
             key=lambda av_dims: (av_dims[0]._as_defn(), av_dims[1])
@@ -3583,6 +3583,15 @@ def __eq__(self, other):
                     or coord_comparison["non_equal_data_dimension"]
                 )
 
+            if result:
+                anc_diff = set(self._ancillary_variables_and_dims) == set(
+                    other._ancillary_variables_and_dims
+                )
+                cm_diff = set(self._cell_measures_and_dims) == set(
+                    other._cell_measures_and_dims
+                )
+                result = anc_diff and cm_diff
+
             # Having checked everything else, check approximate data equality.
             if result:
                 result = da.allclose(
diff --git a/lib/iris/tests/unit/cube/test_Cube.py b/lib/iris/tests/unit/cube/test_Cube.py
index cd2484370b..e1f94be783 100644
--- a/lib/iris/tests/unit/cube/test_Cube.py
+++ b/lib/iris/tests/unit/cube/test_Cube.py
@@ -25,7 +25,13 @@
 from iris.analysis import MEAN
 from iris.aux_factory import HybridHeightFactory
 from iris.cube import Cube
-from iris.coords import AuxCoord, DimCoord, CellMeasure, AncillaryVariable
+from iris.coords import (
+    AuxCoord,
+    DimCoord,
+    CellMeasure,
+    AncillaryVariable,
+    CellMethod,
+)
 from iris.exceptions import (
     CoordinateNotFoundError,
     CellMeasureNotFoundError,
@@ -2206,5 +2212,54 @@ def test_data_bool_not_eq(self):
         self.assertFalse(cube1 == cube2)
 
 
+class Test__eq__meta(tests.IrisTest):
+    def test_ancillary_fail(self):
+        cube1 = Cube([0, 1])
+        cube2 = Cube([0, 1])
+        avr = AncillaryVariable([2, 3], long_name="foo")
+        cube2.add_ancillary_variable(avr, 0)
+        self.assertFalse(cube1 == cube2)
+
+    def test_ancillary_reorder(self):
+        cube1 = Cube([0, 1])
+        cube2 = Cube([0, 1])
+        avr1 = AncillaryVariable([2, 3], long_name="foo")
+        avr2 = AncillaryVariable([4, 5], long_name="bar")
+        # Add the same ancillary variables to cube1 and cube2 in
+        # opposite orders.
+        cube1.add_ancillary_variable(avr1, 0)
+        cube1.add_ancillary_variable(avr2, 0)
+        cube2.add_ancillary_variable(avr2, 0)
+        cube2.add_ancillary_variable(avr1, 0)
+        self.assertTrue(cube1 == cube2)
+
+    def test_cell_measure_fail(self):
+        cube1 = Cube([0, 1])
+        cube2 = Cube([0, 1])
+        cms = CellMeasure([2, 3], measure="area", long_name="foo")
+        cube2.add_cell_measure(cms, 0)
+        self.assertFalse(cube1 == cube2)
+
+    def test_cell_measure_reorder(self):
+        cube1 = Cube([0, 1])
+        cube2 = Cube([0, 1])
+        cms1 = CellMeasure([2, 3], measure="area", long_name="foo")
+        cms2 = CellMeasure([4, 5], measure="area", long_name="bar")
+        # Add the same cell measure to cube1 and cube2 in
+        # opposite orders.
+        cube1.add_cell_measure(cms1, 0)
+        cube1.add_cell_measure(cms2, 0)
+        cube2.add_cell_measure(cms2, 0)
+        cube2.add_cell_measure(cms1, 0)
+        self.assertTrue(cube1 == cube2)
+
+    def test_cell_method_fail(self):
+        cube1 = Cube([0, 1])
+        cube2 = Cube([0, 1])
+        cmth = CellMethod("mean", "time", "6hr")
+        cube2.add_cell_method(cmth)
+        self.assertFalse(cube1 == cube2)
+
+
 if __name__ == "__main__":
     tests.main()