Keep background sub-agent output in the side panel, not the main chat

[pufit]

Problem

Background sub-agents (the Agent tool with run_in_background) spilled all of their tools and thoughts into the main chat instead of their dedicated side-panel. Reloading the page only half-fixed it: already-streamed output snapped into the sidebar, but every subsequent event kept spilling into the main chat.

Root cause

The live streaming handlers routed a sub-agent's child events (those carrying parent_tool_use_id) to a side-panel only if a panel with that id was still status === 'running':

if (parentId && state.panels.some(p => p.id === parentId && p.status === 'running'))

A background sub-agent's Agent tool returns immediately with a task id (like Workflow does). So the immediate tool_result and the backend's immediate subagent_complete marked the panel complete — and scheduled its auto-close — right away. The sub-agent's real activity streams afterward, by which point no panel is running, so each child event fell through into the main chat.

Reload looked correct because the replay path (bufferReplay.ts) enforces a stronger invariant — any parent_tool_use_id event belongs to its panel and never the main chat — but new live events still hit the buggy live path.

Fix

Make the live path honor the same invariant the replay path already uses, and treat background sub-agent panels like the existing Workflow pattern (spawning tool returns immediately → keep the panel open until the detached work actually settles):

• Route any parent_tool_use_id event to its panel by id regardless of status, and never into the main chat — handleThinking / handleToken / handleToolUse / handleToolResult.
• Add a background flag to PanelTab (from input.run_in_background). The immediate Agent result is recorded on the inline chat card, but the panel stays open + running; the premature subagent_complete is ignored; finalizeRunningPanels skips these panels so the launching turn's done doesn't close them.
• Settle background panels via handleBackgroundTasksUpdate once no background task is still running. This is the only reliable completion signal — a detached sub-agent has no per-tool_use_id "done" event (the CLI's task lifecycle is keyed by task_id, which for Agent tasks doesn't carry the spawning tool_use_id). An explicit /stop settles them too.
• Carry the background flag through buffer replay so reconnect-then-live behaves consistently.

Testing

• npm run build (tsc + vite) — clean.
• eslint on the changed files — no new problems (repo's pre-existing lint state unchanged).
• Backend test_streaming.py + test_autonomous_turns.py — 37 passed (no backend changes).
• Manually traced foreground sub-agents, workflows, normal tool calls, and reload-then-live — all unaffected.

Note: there's no frontend test harness in web/ (no vitest/jest), so this is build- and reasoning-verified; the definitive check is a live background-sub-agent run.

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