diff --git a/steward/core/scheduler.py b/steward/core/scheduler.py
index 87bd026..9379304 100644
--- a/steward/core/scheduler.py
+++ b/steward/core/scheduler.py
@@ -42,6 +42,16 @@ class _DueTracker:
             ready.append(task)
         return ready
 
+    def begin(self, tasks: list[ScheduledTask], now: float) -> None:
+        """Start the clock: a task that doesn't run on startup is first due
+        one interval from now. Without this a task that had never run counted
+        from 0 on the loop's clock (host uptime), so every one of them fired
+        on the first tick after a deploy, the hourly cleanup with its Docker
+        log prune among them, holding up agent ingest for minutes (#5431)."""
+        for task in tasks:
+            if not task.run_on_startup:
+                self.last_run.setdefault(task.name, now)
+
     def mark_started(self, task: ScheduledTask, now: float) -> None:
         self.in_flight.add(task.name)
         self.last_run[task.name] = now
@@ -58,10 +68,12 @@ async def start_scheduler(tasks: list[ScheduledTask]) -> None:
         tracker.mark_started(task, now)
         asyncio.create_task(_run_task(task, tracker))
 
+    started = asyncio.get_event_loop().time()
+    tracker.begin(tasks, started)
     for task in tasks:
         if task.run_on_startup:
             logger.info(f"Startup task: {task.name}")
-            _spawn(task, asyncio.get_event_loop().time())
+            _spawn(task, started)
 
     while True:
         now = asyncio.get_event_loop().time()
diff --git a/tests/core/test_scheduler_overlap.py b/tests/core/test_scheduler_overlap.py
index 23e00a3..6b1514b 100644
--- a/tests/core/test_scheduler_overlap.py
+++ b/tests/core/test_scheduler_overlap.py
@@ -56,3 +56,20 @@ def test_independent_tasks_do_not_block_each_other():
     tr = _DueTracker()
     tr.mark_started(a, now=0)             # a hangs
     assert tr.due([a, b], now=10) == [b]  # b still fires
+
+
+def test_a_task_not_run_on_startup_waits_one_interval_after_boot():
+    """#5431: the loop's clock is host uptime, far past every interval. A task
+    marked not to run on startup used to count from 0 and fire on the first
+    tick anyway; it now waits a whole interval from when the loop began."""
+    boot = 1_000_000.0
+    hourly = _task("cleanup", 3600)
+    eager = ScheduledTask(name="poll", coro_factory=lambda: None, interval_seconds=60,
+                          run_on_startup=True)
+    tr = _DueTracker()
+    tr.begin([hourly, eager], now=boot)
+    assert tr.due([hourly], now=boot + 1) == []
+    assert tr.due([hourly], now=boot + 3599) == []
+    assert tr.due([hourly], now=boot + 3600) == [hourly]
+    assert "poll" not in tr.last_run      # started by start_scheduler itself
+