fix(tui): label only detached subagents as background

[kitlangton]

What

Stop foreground running subagents from receiving a misleading Background badge.

Before / After

Before: Any child with structured status: "running" was labeled Background, even while its outer tool call was still running in the foreground. The card therefore claimed the child was backgrounded while the valid ctrl+b hint said it was still blocking.

After: Background appears only when the outer tool call has completed while the child remains running. This is the shared state produced by either ctrl+b detachment or an immediate background: true launch. Foreground and failed calls are not mislabeled.

How

• packages/tui/src/routes/session/index.tsx derives the badge from the outer tool lifecycle plus structured child status and reuses normalized tool input/metadata.
• packages/tui/test/cli/tui/inline-tool-wrap-snapshot.test.tsx covers foreground, detached/async, failed, and completed-child states.

Scope

This does not change subagent execution, backgrounding lifecycle, notification icons, or keyboard behavior. Async-only hint suppression remains covered by #37088, subagent-picker presentation by #36480, and narrow footer action layout by #37180.

Testing

• bun test test/cli/tui/inline-tool-wrap-snapshot.test.tsx (11 pass)
• bun run typecheck from packages/tui
• bun run lint packages/tui/src/routes/session/index.tsx packages/tui/test/cli/tui/inline-tool-wrap-snapshot.test.tsx (0 errors; existing warnings in session/index.tsx)
• Pre-push bun turbo typecheck --concurrency=3 (32/32 tasks)
• Durable reproduction from ses_099291c64ffep0rHSS1Yl4mq42 confirms the reported card was foreground/running despite its false badge
• Matched Effect-native OpenCode Drive runs against the PR base and PR head

Demo

Both recordings use the same Effect program, simulated model responses, 110×34 viewport, and held child-completion gates. Each run shows all three valid lifecycle paths:

1. A foreground child blocks for 2.5 seconds with ctrl+b available.
2. ctrl+b detaches that child, the card becomes Background, and the parent continues before the child is released.
3. A second child launches with background: true, immediately shows Background, and the parent continues before that child is released.

Before

The first foreground child is incorrectly labeled Background before ctrl+b is pressed.

recording-43f39979-7783-4f41-a4ec-2f700c36eef8.mp4

After

The first foreground child has no badge. The badge appears only after ctrl+b, and still appears immediately for the explicit background launch.

recording-700b759a-71f5-4997-8e11-1639adfacf38.mp4

The MP4s were exported with OpenCode Drive #12, which renders the intended ↳ completion icon without the prior missing-glyph artifact.

[kitlangton]

Durable reproduction from the reported local session ses_099291c64ffep0rHSS1Yl4mq42 (next-15652 TUI):

• 11:25:48 EDT: outer subagent tool starts with no background: true input.
• 11:25:55: child session ses_0947733b1ffebLqyQK9mPrbKCi starts.
• 11:27:24: reported screenshot shows a Background badge plus the valid ctrl+b hint, while the outer tool is still running.
• 11:27:45: the outer tool actually completes as background work and queued inputs are admitted.
• 11:27:53: the parent starts its next assistant turn.
• 11:28:14: the child finishes, 21 seconds after the parent resumed.

The persisted tool input has no background flag; its outer state remained running until 11:27:45 while structured child metadata was { status: "running" }. This is the exact false-positive predicate fixed here: child-running metadata must not produce a Background badge until the outer tool call has completed.