diff --git a/ibis/backends/tests/test_impure.py b/ibis/backends/tests/test_impure.py
new file mode 100644
index 0000000000000..b547d51147cbe
--- /dev/null
+++ b/ibis/backends/tests/test_impure.py
@@ -0,0 +1,65 @@
+from __future__ import annotations
+
+import random
+
+import pandas.testing as tm
+import pytest
+
+import ibis
+from ibis import _
+
+
+@ibis.udf.scalar.python(side_effects=True)
+def get_id() -> int:
+    return random.randint(0, 1000)
+
+
+@pytest.mark.parametrize(
+    "impure",
+    [
+        pytest.param(
+            lambda: ibis.random(),
+            id="random",
+        ),
+        pytest.param(
+            lambda: ibis.uuid(),
+            id="uuid",
+        ),
+        pytest.param(
+            get_id,
+            id="udf",
+        ),
+    ],
+)
+def test_impure_uncorrelated(alltypes, impure):
+    df = alltypes.select(x=impure(), y=impure()).execute()
+    assert (df.x != df.y).mean() >= 0.999
+    # Even if the two expressions have the exact same ID, they should still be
+    # uncorrelated
+    common = impure()
+    df = alltypes.select(x=common, y=common).execute()
+    assert (df.x != df.y).mean() >= 0.999
+
+
+@pytest.mark.parametrize(
+    "impure",
+    [
+        pytest.param(
+            lambda: ibis.random(),
+            id="random",
+        ),
+        pytest.param(
+            lambda: ibis.uuid(),
+            id="uuid",
+        ),
+        pytest.param(
+            get_id,
+            id="udf",
+            # once this is fixed, can we unify these params with the above params?
+            marks=pytest.mark.xfail(reason="executed multiple times"),
+        ),
+    ],
+)
+def test_impure_correlated(alltypes, impure):
+    df = alltypes.select(common=impure()).select(x=_.common, y=_.common).execute()
+    tm.assert_series_equal(df.x, df.y, check_names=False)