Relay background-turn responses to Slack

[VojtechBartos]

Problem

When the cloud agent starts a background process and ends its turn to wait for it ("I'll wait for the completion notification"), the wake-up turn that eventually produces the answer never reaches Slack. A background (task-notification) turn resolves no tracked prompt() call, and relayAgentResponse only fires at the three prompt-resolution sites, so the relay for that turn simply never happens. The agent's transcript contains the answer, the user's thread doesn't, and the agent has no way to know the reply was dropped.

Origin: https://posthog.slack.com/archives/C09SK2PAGKF/p1784716116964029

Changes

• agent-server.ts: the ACP transport tap now triggers relayAgentResponse when a clean _posthog/background_turn_complete (stop reason end_turn) passes through. Refusal/error background turns stay silent, matching the tracked path's end_turn gating.
• session-log-writer.ts: relays now consume prose through a new takeUnrelayedAgentResponseParts, which returns only the text blocks no prior relay has delivered. Background turns append to the same turn buffer as the tracked turn before them (only a new prompt resets it), so an unscoped read would re-post already-delivered prose, and a silent wake-up (e.g. the agent just reschedules) would re-post the previous answer verbatim.
• Tracked-turn relays behave exactly as before: the buffer resets at prompt start, so the first take returns the full turn.

Companion backend PR (accepts late relays from completed runs): PostHog/posthog#72897

How did you test this?

Automated only, all in packages/agent:

• New: agent-server.test.ts "background turn relay" — relays prose on a clean background_turn_complete, skips non-end_turn stop reasons, skips silent wake-ups.
• New: session-log-writer.test.ts — take-cursor semantics (only prose since previous take; reset starts over).
• Updated existing relay stubs (question-relay.test.ts, relay-echo tests) to the consuming API.
• Full runs: agent-server.test.ts (150 passed), session-log-writer.test.ts + question-relay.test.ts (87 passed), claude-agent.task-notification.test.ts (2 passed), tsc --noEmit, biome check.

Automatic notifications

• Publish to changelog?
• Alert Sales and Marketing teams?

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[github-actions]

React Doctor found no issues in the changed files. 🎉

_{Reviewed by React Doctor for commit cc759dc.}