feat(web): list background tasks in the session header

The task registry has run every background bash, pwsh, pty-send, and
one-shot subagent since it landed, but only the model could read it: a
human at the Web client could not see that a build was running, tell a
finished task from a stuck one, or find its outcome anywhere but the
`run_in_background` tool card that printed an id and never updated.

Task state now reaches the browser as one whole-snapshot `session/tasks`
mux frame per session, pushed at every registry commit that changes what
that session can see. `TaskService` gains `onTasksChanged`, which is
owner-granular because owner-disposal removal is a change no per-task
record can express. The carrier reads the exact owner the listener hands
it, so a push stays correct while that scope tears down, and reads the
baseline through the non-resuming `ctx.agents.get` so listing never
revives a cold session. The client keeps a last-wins mirror on
`SessionListState`, and a new `dsh-client-ui-task` package renders it
beside the subagent catalog — rendering nothing at all until the session
has a task, so an ordinary conversation grows no new chrome.

Streamed per-task output and human-initiated cancellation are separate
phases; the note records why neither has to undo this channel, and why
no Web path may call the consuming `ctx.tasks.read()`.

packages/client/runtime/README.md

@@ -24,6 +24,8 @@ SlotsService gives the renderer separate bare observables for `useSessions` and
 
 `indexSubagentDescendants()` derives per-parent total and running descendant counts from the retained list mirror. It follows only uninterrupted `origin: 'subagent'` ancestry, so an ordinary fork starts a separate ownership subtree; cycles stop without throwing, and a missing parent remains a harmless key until its summary arrives.
 
+`SessionListState.tasksBySession` mirrors the Host's `session/tasks` frames last-wins, keyed by session and needing no Session instance. An emptied set is stored as an absent key, so absence and `[]` are one representation and consumers never test a sentinel. Two clears keep it from outliving its truth: `session/subscribed` drops the session's mirror, because a fresh generation sends a baseline only for a non-empty set and a retained list would survive as a phantom, and `host/session-removed` drops it again, because owner disposal removed the records on the mux stream while the removal frame rides the host stream, leaving the two with no relative order.
+
 `SessionsService.search(query, signal)` is a stateless one-shot action over the `session.search` RPC. It returns ranked session/snippet pairs without putting query, loading, or error state into the shared Session list, so each UI owner controls debounce, cancellation, stale-response suppression, and fallback presentation. `searchResultLimit` re-exposes `SESSION_SEARCH_RESULT_LIMIT` — the bound the response schema itself enforces — as injected presentation data, so client plugins do not duplicate it. It is a protocol constant rather than per-connection state, so the connection handle does not carry it.
 
 ## New Session and the blank mirror