diff --git a/modules/mesh-build-image/cmd/mesh-build-image/module.go b/modules/mesh-build-image/cmd/mesh-build-image/module.go
index 819654c1..4fb2a146 100644
--- a/modules/mesh-build-image/cmd/mesh-build-image/module.go
+++ b/modules/mesh-build-image/cmd/mesh-build-image/module.go
@@ -2,6 +2,7 @@ package main
 
 import (
 	"context"
+	"errors"
 	"fmt"
 	"net/http"
 	"sync"
@@ -97,11 +98,25 @@ func (m *Module) look(ctx context.Context) (Look, error) {
 		return l, &buildstep.Refusal{Text: err.Error(), Have: l.Buildah}
 	}
 	if err := m.tunDevice(); err != nil {
-		return l, &buildstep.Refusal{Text: err.Error(), Have: l.Buildah}
+		return l, &notEligible{&buildstep.Refusal{Text: err.Error(), Have: l.Buildah}}
 	}
 	return l, nil
 }
 
+// notEligible is a refusal of a machine that runs the module fine but cannot build here now (no tun device: its asks go
+// to another node). It is a *buildstep.Refusal for a step, and the health stays healthy for it (issue 1126): the
+// assigner reads unhealthy as a failed delivery and holds the build for every node. What cannot run (buildah missing,
+// no ranges, no namespace, no runtime directory, no room) stays unhealthy.
+type notEligible struct{ *buildstep.Refusal }
+
+func (e *notEligible) Unwrap() error { return e.Refusal }
+
+// isNotEligible says whether err is a refusal that leaves the module healthy.
+func isNotEligible(err error) bool {
+	var n *notEligible
+	return errors.As(err, &n)
+}
+
 // Check is the guard: nil when the machine can build, else the refusal. Every look is kept for Eligible, and a change
 // wakes the puller, so a node that can build again takes asks at once.
 func (m *Module) Check(ctx context.Context) error {
@@ -120,6 +135,13 @@ func (m *Module) keep(err error) {
 	changed := m.last != why
 	m.last = why
 	m.lastMu.Unlock()
+	if changed && m.Say != nil {
+		if why == "" {
+			m.Say("this machine is eligible to build images again")
+		} else {
+			m.Say("this machine takes no image build now: %s", why)
+		}
+	}
 	if changed && m.Held != nil {
 		m.Held.Wake()
 	}
@@ -155,12 +177,16 @@ func (m *Module) Start(ctx context.Context) {
 func (m *Module) Health(ctx context.Context) health.Answer {
 	l, err := m.look(ctx)
 	m.keep(err)
-	if err != nil {
+	if err != nil && !isNotEligible(err) {
 		return health.Unhealthy(err.Error())
 	}
 	if err := m.room(); err != nil {
 		return health.Unhealthy(err.Error())
 	}
+	if err != nil {
+		// Not eligible: the module runs, it takes no image ask here now, and says why; the capacity record says it too.
+		return health.Healthy("not eligible to build images, so it takes no image ask: " + err.Error())
+	}
 	return health.Healthy(fmt.Sprintf("buildah %s installed; %s holds %d subordinate uids and %d gids; a user namespace can be made; %s opens", l.Buildah, l.Account, l.UIDs, l.GIDs, m.tunPath()))
 }
 
@@ -168,7 +194,7 @@ func (m *Module) Health(ctx context.Context) health.Answer {
 func (m *Module) Status(ctx context.Context) map[string]any {
 	l, err := m.look(ctx)
 	out := map[string]any{"node": m.Node, "account": m.Account, "buildah": l.Buildah, "subuids": l.UIDs, "subgids": l.GIDs, "refusing": err != nil,
-		"cpu": m.cpuStatus()}
+		"eligible": err == nil, "cpu": m.cpuStatus()}
 	if err != nil {
 		out["why"] = err.Error()
 	}
diff --git a/modules/mesh-build-image/cmd/mesh-build-image/module_test.go b/modules/mesh-build-image/cmd/mesh-build-image/module_test.go
index cd26685b..fe2c5f2f 100644
--- a/modules/mesh-build-image/cmd/mesh-build-image/module_test.go
+++ b/modules/mesh-build-image/cmd/mesh-build-image/module_test.go
@@ -2,7 +2,9 @@ package main
 
 import (
 	"context"
+	"fmt"
 	"os"
+	"path/filepath"
 	"strings"
 	"testing"
 
@@ -54,7 +56,6 @@ func TestHealthIsUnhealthyWithoutWhatABuildNeeds(t *testing.T) {
 	}{
 		{"subuid", func(m *Module, p string) { m.SubUIDFile = p }, []string{"/subuid", "mesh-build-image"}},
 		{"subgid", func(m *Module, p string) { m.SubGIDFile = p }, []string{"/subgid", "mesh-build-image"}},
-		{"tun", func(m *Module, p string) { m.Tun = p + "/dev/net/tun" }, []string{"not eligible: ", "/dev/net/tun cannot be opened", "tun"}},
 	} {
 		f := newFake(t)
 		m := moduleWith(t, f, newStates())
@@ -74,6 +75,61 @@ func TestHealthIsUnhealthyWithoutWhatABuildNeeds(t *testing.T) {
 	}
 }
 
+// 1126: a node that cannot build here now (no tun device) stays healthy, saying "not eligible" and why; the assigner reads
+// unhealthy as a failed delivery and held the new build for every node. The eligibility, the status and the log say it too.
+func TestANodeWithoutTunIsHealthyAndSaysItIsNotEligible(t *testing.T) {
+	f := newFake(t)
+	m := moduleWith(t, f, newStates())
+	var said []string
+	m.Say = func(format string, a ...any) { said = append(said, fmt.Sprintf(format, a...)) }
+	m.Tun = t.TempDir() + "/dev/net/tun"
+	a := m.Health(context.Background())
+	if a.State != health.StateHealthy || !a.Healthy {
+		t.Fatalf("health = %+v, want healthy", a)
+	}
+	for _, want := range []string{"not eligible", "/dev/net/tun cannot be opened", "tun"} {
+		if !strings.Contains(a.Why, want) {
+			t.Errorf("health says %q, lacks %q", a.Why, want)
+		}
+	}
+	if got := m.Eligible(); !strings.Contains(got, "not eligible: ") {
+		t.Errorf("eligible says %q", got)
+	}
+	if s := m.Status(context.Background()); s["eligible"] != false || s["refusing"] != true {
+		t.Errorf("status = %v, want not eligible", s)
+	}
+	// One log line for the change, none for the same look again; one when it can build again.
+	m.Health(context.Background())
+	if len(said) != 1 || !strings.Contains(said[0], "takes no image build now") {
+		t.Errorf("said = %q, want one line for the change", said)
+	}
+	m.Tun = filepath.Join(t.TempDir(), "tun")
+	os.WriteFile(m.Tun, nil, 0o644)
+	if a := m.Health(context.Background()); !a.Healthy || strings.Contains(a.Why, "not eligible") {
+		t.Errorf("health = %+v, want plain healthy", a)
+	}
+	if len(said) != 2 || !strings.Contains(said[1], "eligible to build images again") {
+		t.Errorf("said = %q, want the return said", said)
+	}
+	// A step still ends refused with the cause named.
+	m.Tun = t.TempDir() + "/none"
+	if err := m.Check(context.Background()); err == nil || !strings.Contains(err.Error(), "not eligible") {
+		t.Errorf("check = %v", err)
+	}
+}
+
+// 1126: what cannot run stays unhealthy even beside a missing tun: buildah missing is not "not eligible".
+func TestWithoutBuildahTheNodeIsUnhealthyNotIneligible(t *testing.T) {
+	m := moduleWith(t, newFake(t), newStates())
+	m.Tun = t.TempDir() + "/none"
+	m.Run = func(ctx context.Context, env []string, name string, args ...string) forgekey.Ran {
+		return forgekey.Ran{Status: 127, Err: "exec: \"buildah\": executable file not found in $PATH"}
+	}
+	if a := m.Health(context.Background()); a.Healthy {
+		t.Errorf("health = %+v, want unhealthy", a)
+	}
+}
+
 func TestHealthIsUnhealthyWhenANamespaceCannotBeMade(t *testing.T) {
 	f := newFake(t)
 	m := moduleWith(t, f, newStates())
diff --git a/modules/mesh-build-image/cmd/mesh-build-image/tools.go b/modules/mesh-build-image/cmd/mesh-build-image/tools.go
index 8455637c..6bd15a55 100644
--- a/modules/mesh-build-image/cmd/mesh-build-image/tools.go
+++ b/modules/mesh-build-image/cmd/mesh-build-image/tools.go
@@ -54,7 +54,7 @@ func (m *Module) Tools(steps []StepTool) []stdio.Tool {
 			Run: func(map[string]any) (any, error) { return m.Status(m.Ctx), nil }},
 		stdio.Tool{Name: HealthTool,
 			Description: "How the image build module is: unhealthy while buildah cannot be run, while its account holds no " +
-				"subordinate user or group range, or while a user namespace cannot be made, saying which; healthy otherwise. Read-only.",
+				"subordinate user or group range, or while a user namespace cannot be made, saying which; healthy otherwise. A machine that cannot build here now (no tun device) stays healthy and says in its words \"not eligible\", with the cause; it takes no image ask meanwhile (issue 1126). Read-only.",
 			Run: func(map[string]any) (any, error) { return m.Health(m.Ctx), nil }},
 	)
 }