diff --git a/newsfragments/890.feature.rst b/newsfragments/890.feature.rst
new file mode 100644
index 0000000000..e274f6982c
--- /dev/null
+++ b/newsfragments/890.feature.rst
@@ -0,0 +1,3 @@
+Added an internal mechanism for pytest-trio's
+`Hypothesis <https://hypothesis.readthedocs.io>`__ integration
+to make the task scheduler reproducible and avoid flaky tests.
diff --git a/trio/_core/_run.py b/trio/_core/_run.py
index f96f6c9181..98b44039b8 100644
--- a/trio/_core/_run.py
+++ b/trio/_core/_run.py
@@ -1,4 +1,5 @@
 import functools
+import itertools
 import logging
 import os
 import random
@@ -58,6 +59,12 @@
 else:  # pragma: no cover
     raise NotImplementedError("unsupported platform")
 
+# When running under Hypothesis, we want examples to be reproducible and
+# shrinkable.  pytest-trio's Hypothesis integration monkeypatches this
+# variable to True, and registers the Random instance _r for Hypothesis
+# to manage for each test case, which together should make Trio's task
+# scheduling loop deterministic.  We have a test for that, of course.
+_ALLOW_DETERMINISTIC_SCHEDULING = False
 _r = random.Random()
 
 # Used to log exceptions in instruments
@@ -682,6 +689,7 @@ class Task:
     name = attr.ib()
     # PEP 567 contextvars context
     context = attr.ib()
+    _counter = attr.ib(init=False, factory=itertools.count().__next__)
 
     # Invariant:
     # - for unscheduled tasks, _next_send is None
@@ -1557,6 +1565,12 @@ def run_impl(runner, async_fn, args):
         # change too, like the deadlines tie-breaker and the non-deterministic
         # ordering of task._notify_queues.)
         batch = list(runner.runq)
+        if _ALLOW_DETERMINISTIC_SCHEDULING:
+            # We're running under Hypothesis, and pytest-trio has patched this
+            # in to make the scheduler deterministic and avoid flaky tests.
+            # It's not worth the (small) performance cost in normal operation,
+            # since we'll shuffle the list and _r is only seeded for tests.
+            batch.sort(key=lambda t: t._counter)
         runner.runq.clear()
         _r.shuffle(batch)
         while batch:
diff --git a/trio/tests/test_scheduler_determinism.py b/trio/tests/test_scheduler_determinism.py
new file mode 100644
index 0000000000..ba5f469396
--- /dev/null
+++ b/trio/tests/test_scheduler_determinism.py
@@ -0,0 +1,42 @@
+import trio
+
+
+async def scheduler_trace():
+    """Returns a scheduler-dependent value we can use to check determinism."""
+    trace = []
+
+    async def tracer(name):
+        for i in range(10):
+            trace.append((name, i))
+            await trio.sleep(0)
+
+    async with trio.open_nursery() as nursery:
+        for i in range(5):
+            nursery.start_soon(tracer, i)
+
+    return tuple(trace)
+
+
+def test_the_trio_scheduler_is_not_deterministic():
+    # At least, not yet.  See https://github.com/python-trio/trio/issues/32
+    traces = []
+    for _ in range(10):
+        traces.append(trio.run(scheduler_trace))
+    assert len(set(traces)) == len(traces)
+
+
+def test_the_trio_scheduler_is_deterministic_if_seeded(monkeypatch):
+    monkeypatch.setattr(
+        trio._core._run, "_ALLOW_DETERMINISTIC_SCHEDULING", True
+    )
+    traces = []
+    for _ in range(10):
+        state = trio._core._run._r.getstate()
+        try:
+            trio._core._run._r.seed(0)
+            traces.append(trio.run(scheduler_trace))
+        finally:
+            trio._core._run._r.setstate(state)
+
+    assert len(traces) == 10
+    assert len(set(traces)) == 1