diff --git a/properties/test_index_manipulation.py b/properties/test_index_manipulation.py
index e04db83fd84..7058bc248c8 100644
--- a/properties/test_index_manipulation.py
+++ b/properties/test_index_manipulation.py
@@ -24,21 +24,9 @@
 
 import xarray.testing.strategies as xrst
 
-
-@st.composite
-def unique(draw, strategy):
-    # https://stackoverflow.com/questions/73737073/create-hypothesis-strategy-that-returns-unique-values
-    seen = draw(st.shared(st.builds(set), key="key-for-unique-elems"))
-    return draw(
-        strategy.filter(lambda x: x not in seen).map(lambda x: seen.add(x) or x)
-    )
-
-
-# Share to ensure we get unique names on each draw,
-# so we don't try to add two variables with the same name
-# or stack to a dimension with a name that already exists in the Dataset.
-UNIQUE_NAME = unique(strategy=xrst.names())
-DIM_NAME = xrst.dimension_names(name_strategy=UNIQUE_NAME, min_dims=1, max_dims=1)
+# Strategy for generating names - uniqueness is enforced by the state machine
+NAME_STRATEGY = xrst.names()
+DIM_NAME = xrst.dimension_names(name_strategy=NAME_STRATEGY, min_dims=1, max_dims=1)
 index_variables = st.builds(
     xr.Variable,
     data=npst.arrays(
@@ -75,25 +63,45 @@ def __init__(self):
         self.indexed_dims = []
         self.multi_indexed_dims = []
 
-    @initialize(var=index_variables)
-    def init_ds(self, var):
+        # Track all used names to ensure uniqueness (avoids flaky Hypothesis tests)
+        self.used_names: set[str] = set()
+
+    def _draw_unique_name(self, data) -> str:
+        """Draw a name that hasn't been used yet in this test case."""
+        name = data.draw(NAME_STRATEGY.filter(lambda x: x not in self.used_names))
+        self.used_names.add(name)
+        return name
+
+    def _draw_unique_var(self, data) -> xr.Variable:
+        """Draw an index variable with a unique dimension name."""
+        var = data.draw(index_variables)
+        # Replace with a guaranteed unique name
+        new_name = self._draw_unique_name(data)
+        return xr.Variable(dims=(new_name,), data=var.data, attrs=var.attrs)
+
+    @initialize(data=st.data())
+    def init_ds(self, data):
         """Initialize the Dataset so that at least one rule will always fire."""
+        var = self._draw_unique_var(data)
         (name,) = var.dims
+        note(f"initializing with dimension coordinate {name}")
         add_dim_coord_and_data_var(self.dataset, var)
 
         self.indexed_dims.append(name)
 
     # TODO: stacking with a timedelta64 index and unstacking converts it to object
-    @rule(var=index_variables)
-    def add_dim_coord(self, var):
+    @rule(data=st.data())
+    def add_dim_coord(self, data):
+        var = self._draw_unique_var(data)
         (name,) = var.dims
         note(f"adding dimension coordinate {name}")
         add_dim_coord_and_data_var(self.dataset, var)
 
         self.indexed_dims.append(name)
 
-    @rule(var=index_variables)
-    def assign_coords(self, var):
+    @rule(data=st.data())
+    def assign_coords(self, data):
+        var = self._draw_unique_var(data)
         (name,) = var.dims
         note(f"assign_coords: {name}")
         self.dataset = self.dataset.assign_coords({name: var})
@@ -117,9 +125,10 @@ def reset_index(self, data):
         elif dim in self.multi_indexed_dims:
             del self.multi_indexed_dims[self.multi_indexed_dims.index(dim)]
 
-    @rule(newname=UNIQUE_NAME, data=st.data(), create_index=st.booleans())
+    @rule(data=st.data(), create_index=st.booleans())
     @precondition(lambda self: bool(self.indexed_dims))
-    def stack(self, newname, data, create_index):
+    def stack(self, data, create_index):
+        newname = self._draw_unique_name(data)
         oldnames = data.draw(
             st.lists(
                 st.sampled_from(self.indexed_dims),
@@ -158,9 +167,10 @@ def unstack(self, data):
             # TODO: fix this
             pass
 
-    @rule(newname=UNIQUE_NAME, data=st.data())
+    @rule(data=st.data())
     @precondition(lambda self: bool(self.dataset.variables))
-    def rename_vars(self, newname, data):
+    def rename_vars(self, data):
+        newname = self._draw_unique_name(data)
         dim = data.draw(st.sampled_from(sorted(self.dataset.variables)))
         # benbovy: "skip the default indexes invariant test when the name of an
         # existing dimension coordinate is passed as input kwarg or dict key