diff --git a/src/main/claude/claude-background-task-tracker.test.ts b/src/main/claude/claude-background-task-tracker.test.ts
index afac5a1a23f..7d17ee086ca 100644
--- a/src/main/claude/claude-background-task-tracker.test.ts
+++ b/src/main/claude/claude-background-task-tracker.test.ts
@@ -55,8 +55,8 @@ describe('ClaudeBackgroundTaskTracker', () => {
     expect(tracker.state).toEqual({
       state: 'monitoring',
       tasks: [
-        { id: 'agent-1', kind: 'agent' },
-        { id: 'agent-2', kind: 'agent' }
+        { id: 'agent-1', kind: 'agent', stoppable: false },
+        { id: 'agent-2', kind: 'agent', stoppable: false }
       ]
     })
     // The turn IS the outcome of work the provider marked foreground.
@@ -66,6 +66,29 @@ describe('ClaudeBackgroundTaskTracker', () => {
     expect(tracker.stoppableTaskIds).toEqual([])
   })
 
+  it('marks a foreground row not stoppable and leaves a backgrounded row alone', () => {
+    // `stopTask` has no foreground target, so the row must not offer a Stop that
+    // would silently do nothing. A backgrounded row stays untouched on the wire.
+    const tracker = new ClaudeBackgroundTaskTracker()
+    tracker.observe({ type: 'user' }, true)
+    tracker.observe(
+      system('task_started', {
+        task_id: 'fore-1',
+        task_type: 'local_agent',
+        is_backgrounded: false
+      })
+    )
+    tracker.observe(
+      system('task_started', { task_id: 'back-1', task_type: 'local_agent', is_backgrounded: true })
+    )
+
+    expect(tracker.state?.tasks).toEqual([
+      { id: 'fore-1', kind: 'agent', stoppable: false },
+      { id: 'back-1', kind: 'agent' }
+    ])
+    expect(tracker.stoppableTaskIds).toEqual(['back-1'])
+  })
+
   it('uses an explicit background update for a foreground task and ignores progress alone', () => {
     const tracker = new ClaudeBackgroundTaskTracker()
     tracker.observe({ type: 'user' }, true)
@@ -80,7 +103,7 @@ describe('ClaudeBackgroundTaskTracker', () => {
       tracker.observe(system('task_progress', { task_id: 'task-1', description: 'still working' }))
     ).toBe(false)
     // Live while the turn runs, then retired by that turn's `result`.
-    expect(tracker.state?.tasks).toEqual([{ id: 'task-1', kind: 'command' }])
+    expect(tracker.state?.tasks).toEqual([{ id: 'task-1', kind: 'command', stoppable: false }])
     tracker.observe(result())
     expect(tracker.state).toBeNull()
 
diff --git a/src/main/claude/claude-background-task-tracker.ts b/src/main/claude/claude-background-task-tracker.ts
index 7667f5c27c3..b5bd6e13d26 100644
--- a/src/main/claude/claude-background-task-tracker.ts
+++ b/src/main/claude/claude-background-task-tracker.ts
@@ -265,6 +265,9 @@ export class ClaudeBackgroundTaskTracker {
       details.push({
         id,
         kind: task.kind,
+        // Foreground work is not a target `stopTask` accepts, so the row says so
+        // rather than drawing a Stop that would silently do nothing.
+        ...(task.backgrounded ? {} : { stoppable: false }),
         ...(task.description ? { description: task.description } : {})
       })
     }
diff --git a/src/renderer/src/components/native-chat/NativeChatBackgroundTasksStatus.test.tsx b/src/renderer/src/components/native-chat/NativeChatBackgroundTasksStatus.test.tsx
index f038bb1fcd9..435012539c5 100644
--- a/src/renderer/src/components/native-chat/NativeChatBackgroundTasksStatus.test.tsx
+++ b/src/renderer/src/components/native-chat/NativeChatBackgroundTasksStatus.test.tsx
@@ -80,9 +80,7 @@ describe('NativeChatBackgroundTasksStatus row glyphs', () => {
     const monitorGlyph = glyphClassFor('monitor')
     cleanup()
     const { container } = render(<AgentStateDot state="monitoring" size="md" title={null} />)
-    const dotGlyph = lucideGlyphName(
-      container.querySelector('svg')?.getAttribute('class') ?? ''
-    )
+    const dotGlyph = lucideGlyphName(container.querySelector('svg')?.getAttribute('class') ?? '')
     expect(monitorGlyph).not.toBe('')
     expect(monitorGlyph).toBe(dotGlyph)
   })
@@ -122,6 +120,30 @@ describe('NativeChatBackgroundTasksStatus stop affordances', () => {
     expect(screen.getByLabelText('Stop background tasks')).toBeInTheDocument()
   })
 
+  it('withholds a row stop the host reported it cannot act on', () => {
+    // Claude publishes foreground rows with `stoppable: false`: the session
+    // accepts targeted stops, but not for this row.
+    const onStop = vi.fn()
+    render(
+      <NativeChatBackgroundTasksStatus
+        tasks={[
+          { id: 'fore-1', kind: 'agent', description: 'in-turn subagent', stoppable: false },
+          { id: 'back-1', kind: 'agent', description: 'backgrounded subagent' }
+        ]}
+        supportsTaskStop
+        supportsStopAll
+        stoppingTaskIds={new Set()}
+        stoppingAll={false}
+        onStop={onStop}
+      />
+    )
+    fireEvent.click(screen.getByRole('button', { expanded: false }))
+
+    expect(screen.getByText('in-turn subagent')).toBeInTheDocument()
+    expect(screen.queryByLabelText('Stop in-turn subagent')).not.toBeInTheDocument()
+    expect(screen.getByLabelText('Stop backgrounded subagent')).toBeInTheDocument()
+  })
+
   it('offers no stop at all when the provider exposes none', () => {
     // Codex: a Stop button here would be a control that cannot act.
     renderStrip({ supportsTaskStop: false, supportsStopAll: false })
diff --git a/src/renderer/src/components/native-chat/NativeChatBackgroundTasksStatus.tsx b/src/renderer/src/components/native-chat/NativeChatBackgroundTasksStatus.tsx
index 9d3e899a359..4a3f968d512 100644
--- a/src/renderer/src/components/native-chat/NativeChatBackgroundTasksStatus.tsx
+++ b/src/renderer/src/components/native-chat/NativeChatBackgroundTasksStatus.tsx
@@ -13,8 +13,17 @@ function backgroundTaskLabel(task: AgentSessionBackgroundTask): string {
   return task.description || backgroundTaskKindLabel(task.kind)
 }
 
+/** Absent means stoppable: a host that predates the field published only rows
+ *  its stop could act on. Only an explicit `false` withholds the control. */
+function backgroundTaskStoppable(task: AgentSessionBackgroundTask): boolean {
+  return task.stoppable !== false
+}
+
 export function NativeChatBackgroundTasksStatus(props: {
   tasks: readonly AgentSessionBackgroundTask[]
+  /** Whether the host accepts a targeted stop at all. A row still withholds its
+   *  own button when the row itself reports it cannot be stopped — same rule as
+   *  `supportsStopAll`: never a button that cannot act. */
   supportsTaskStop: boolean
   /** False when the provider exposes no honest stop at all; the fallback
    *  control is hidden rather than offering a button that cannot act. */
@@ -75,12 +84,9 @@ export function NativeChatBackgroundTasksStatus(props: {
                       key={task.id}
                       className="flex min-w-0 items-center gap-2 text-foreground/80"
                     >
-                      <BackgroundTaskKindIcon
-                        kind={task.kind}
-                        className="text-muted-foreground"
-                      />
+                      <BackgroundTaskKindIcon kind={task.kind} className="text-muted-foreground" />
                       <span className="min-w-0 flex-1 break-words">{label}</span>
-                      {props.supportsTaskStop ? (
+                      {props.supportsTaskStop && backgroundTaskStoppable(task) ? (
                         <Button
                           type="button"
                           variant="ghost"
diff --git a/src/shared/agent-session-wire.ts b/src/shared/agent-session-wire.ts
index 2b5c0e5ae07..6ac9f9a8228 100644
--- a/src/shared/agent-session-wire.ts
+++ b/src/shared/agent-session-wire.ts
@@ -60,6 +60,11 @@ export type AgentSessionBackgroundTask = {
   id: string
   kind: 'agent' | 'workflow' | 'command' | 'monitor' | 'unknown'
   description?: string
+  /** Whether this row's own stop can act on it. Absent means yes: every host
+   *  that predates this field published only backgrounded, stoppable rows, and
+   *  a client that read absence as "not stoppable" would hide a working control
+   *  on those hosts. A row the host cannot target sends `false`. */
+  stoppable?: boolean
 }
 
 export type AgentSessionBackgroundTaskState = {
diff --git a/src/shared/structured-agent-session-reducer.test.ts b/src/shared/structured-agent-session-reducer.test.ts
index 9aec45c8bf1..2ec9bae8596 100644
--- a/src/shared/structured-agent-session-reducer.test.ts
+++ b/src/shared/structured-agent-session-reducer.test.ts
@@ -87,6 +87,46 @@ describe('structured agent session reducer', () => {
     expect(updated.items).toBe(initial.items)
   })
 
+  it("republishes when only a row's stoppability changes", () => {
+    // A row losing its stop is the whole difference between an honest control
+    // and a dead one, so it must not be dropped as an equal state.
+    const backgroundTasks = {
+      state: 'monitoring' as const,
+      supportsTaskStop: true,
+      tasks: [{ id: 'task-1', kind: 'agent' as const }]
+    }
+    const initial = reduceStructuredAgentSession(EMPTY_STRUCTURED_AGENT_SESSION, {
+      type: 'event',
+      event: {
+        type: 'snapshot',
+        sessionId: 'session-a',
+        fence: 1,
+        page: { ...hydrationPage([]), backgroundTasks }
+      }
+    })
+    const updated = reduceStructuredAgentSession(initial, {
+      type: 'event',
+      event: {
+        type: 'batch',
+        sessionId: 'session-a',
+        batch: {
+          cursor: { epoch: 'epoch-a', sequence: 0 },
+          items: [],
+          removedItemIds: [],
+          submissions: []
+        },
+        backgroundTasks: {
+          ...backgroundTasks,
+          tasks: [{ id: 'task-1', kind: 'agent' as const, stoppable: false }]
+        }
+      }
+    })
+
+    expect(updated.backgroundTasks?.tasks).toEqual([
+      { id: 'task-1', kind: 'agent', stoppable: false }
+    ])
+  })
+
   it('uses the bounded hydration page pagination boundary', () => {
     const restored = reduceStructuredAgentSession(EMPTY_STRUCTURED_AGENT_SESSION, {
       type: 'event',
diff --git a/src/shared/structured-agent-session-reducer.ts b/src/shared/structured-agent-session-reducer.ts
index 10897c3f61f..16b8814648d 100644
--- a/src/shared/structured-agent-session-reducer.ts
+++ b/src/shared/structured-agent-session-reducer.ts
@@ -74,7 +74,8 @@ function backgroundTaskStatesEqual(
     (task, index) =>
       task.id === right.tasks?.[index]?.id &&
       task.kind === right.tasks[index]?.kind &&
-      task.description === right.tasks[index]?.description
+      task.description === right.tasks[index]?.description &&
+      task.stoppable === right.tasks[index]?.stoppable
   )
 }