diff --git a/nomad/core_sched_test.go b/nomad/core_sched_test.go
index a075b6377ce4..6ecb078fa487 100644
--- a/nomad/core_sched_test.go
+++ b/nomad/core_sched_test.go
@@ -2394,3 +2394,64 @@ func TestCoreScheduler_CSIVolumeClaimGC(t *testing.T) {
 	}, time.Second*1, 10*time.Millisecond, "claims were released unexpectedly")
 
 }
+
+func TestCoreScheduler_FailLoop(t *testing.T) {
+	t.Parallel()
+	require := require.New(t)
+
+	srv, cleanupSrv := TestServer(t, func(c *Config) {
+		c.NumSchedulers = 0 // Prevent automatic dequeue
+		c.EvalDeliveryLimit = 2
+		c.EvalFailedFollowupBaselineDelay = time.Duration(50 * time.Millisecond)
+		c.EvalFailedFollowupDelayRange = time.Duration(1 * time.Millisecond)
+	})
+	defer cleanupSrv()
+	codec := rpcClient(t, srv)
+	sched := []string{structs.JobTypeCore}
+
+	testutil.WaitForResult(func() (bool, error) {
+		return srv.evalBroker.Enabled(), nil
+	}, func(err error) {
+		t.Fatalf("should enable eval broker")
+	})
+
+	// Enqueue a core job eval that can never succeed because it was enqueued
+	// by another leader that's now gone
+	expected := srv.coreJobEval(structs.CoreJobCSIPluginGC, 100)
+	expected.LeaderACL = "nonsense"
+	srv.evalBroker.Enqueue(expected)
+
+	nack := func(evalID, token string) error {
+		req := &structs.EvalAckRequest{
+			EvalID:       evalID,
+			Token:        token,
+			WriteRequest: structs.WriteRequest{Region: "global"},
+		}
+		var resp structs.GenericResponse
+		return msgpackrpc.CallWithCodec(codec, "Eval.Nack", req, &resp)
+	}
+
+	out, token, err := srv.evalBroker.Dequeue(sched, time.Second*5)
+	require.NoError(err)
+	require.NotNil(out)
+	require.Equal(expected, out)
+
+	// first fail
+	require.NoError(nack(out.ID, token))
+
+	out, token, err = srv.evalBroker.Dequeue(sched, time.Second*5)
+	require.NoError(err)
+	require.NotNil(out)
+	require.Equal(expected, out)
+
+	// second fail, should not result in failed-follow-up
+	require.NoError(nack(out.ID, token))
+
+	out, token, err = srv.evalBroker.Dequeue(sched, time.Second*5)
+	require.NoError(err)
+	if out != nil {
+		t.Fatalf(
+			"failed core jobs should not result in follow-up. TriggeredBy: %v",
+			out.TriggeredBy)
+	}
+}
diff --git a/nomad/leader.go b/nomad/leader.go
index c19b2159b261..8ad929b181eb 100644
--- a/nomad/leader.go
+++ b/nomad/leader.go
@@ -640,25 +640,31 @@ func (s *Server) reapFailedEvaluations(stopCh chan struct{}) {
 			updateEval.StatusDescription = fmt.Sprintf("evaluation reached delivery limit (%d)", s.config.EvalDeliveryLimit)
 			s.logger.Warn("eval reached delivery limit, marking as failed", "eval", updateEval.GoString())
 
-			// Create a follow-up evaluation that will be used to retry the
-			// scheduling for the job after the cluster is hopefully more stable
-			// due to the fairly large backoff.
-			followupEvalWait := s.config.EvalFailedFollowupBaselineDelay +
-				time.Duration(rand.Int63n(int64(s.config.EvalFailedFollowupDelayRange)))
-
-			followupEval := eval.CreateFailedFollowUpEval(followupEvalWait)
-			updateEval.NextEval = followupEval.ID
-			updateEval.UpdateModifyTime()
-
-			// Update via Raft
-			req := structs.EvalUpdateRequest{
-				Evals: []*structs.Evaluation{updateEval, followupEval},
-			}
-			if _, _, err := s.raftApply(structs.EvalUpdateRequestType, &req); err != nil {
-				s.logger.Error("failed to update failed eval and create a follow-up", "eval", updateEval.GoString(), "error", err)
-				continue
+			// Core job evals that fail or span leader elections will never
+			// succeed because the follow-up doesn't have the leader ACL. We
+			// rely on the leader to schedule new core jobs periodically
+			// instead.
+			if eval.Type != structs.JobTypeCore {
+
+				// Create a follow-up evaluation that will be used to retry the
+				// scheduling for the job after the cluster is hopefully more stable
+				// due to the fairly large backoff.
+				followupEvalWait := s.config.EvalFailedFollowupBaselineDelay +
+					time.Duration(rand.Int63n(int64(s.config.EvalFailedFollowupDelayRange)))
+
+				followupEval := eval.CreateFailedFollowUpEval(followupEvalWait)
+				updateEval.NextEval = followupEval.ID
+				updateEval.UpdateModifyTime()
+
+				// Update via Raft
+				req := structs.EvalUpdateRequest{
+					Evals: []*structs.Evaluation{updateEval, followupEval},
+				}
+				if _, _, err := s.raftApply(structs.EvalUpdateRequestType, &req); err != nil {
+					s.logger.Error("failed to update failed eval and create a follow-up", "eval", updateEval.GoString(), "error", err)
+					continue
+				}
 			}
-
 			// Ack completion
 			s.evalBroker.Ack(eval.ID, token)
 		}