feat: live agent answers (SSE streaming) + «Чат» tab on phones

- POST /dialog/{token}/events/stream: Server-Sent Events with the agent's steps («Смотрю вашу карту ЕМИАС…»,
  «Ищу в клинических рекомендациях…»), the answer as the model writes it (OpenRouter stream=true), then the
  final replies (same as /events, incl. the forbidden-output filter and sources). X-Accel-Buffering: no.
- Web: free-text messages use the stream — a live bubble with steps, typing dots and a cursor; falls back to
  the plain request if the stream breaks.
- Phones: the bottom tab bar gets «Чат» (it had no way back to the chat).

Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>

app/api/routes.py

@@ -8,9 +8,14 @@ disclosed — the dashboard polls every 2–3 s), `ack_alert` (in `repo.ack_aler
 `app/api/fixtures.py` is kept for tests and as the demo cabinet reference (doctor id).
 """
 
+import asyncio
+import json
+import logging
 import secrets
+from typing import Any
 
 from fastapi import APIRouter, HTTPException
+from fastapi.responses import StreamingResponse
 from sqlalchemy import select
 
 from app.api import demo as console
@@ -42,10 +47,12 @@ from app.db.identity import Patient
 from app.db.session import get_clinical_session, get_identity_session
 from app.dialog.contracts import Event, get_engine
 from app.integrations.base import integrations
+from app.llm.agent import stream as agent_stream
 from app.scheduler import jobs as scheduler
 from app.services.anonymous_care import display_name as anonymous_name
 from app.services.monitoring import refresh_status
 
+log = logging.getLogger(__name__)
 router = APIRouter(prefix="/api/v1")
 router.include_router(max_notifications.router)
 router.include_router(questionnaire_router)
@@ -219,6 +226,49 @@ async def dialog_event(token: str, body: DialogIn) -> DialogOut:
     return DialogOut(replies=await get_engine().handle(token, body))
 
 
+@router.post("/dialog/{token}/events/stream", tags=["dialog"])
+async def dialog_event_stream(token: str, body: DialogIn) -> StreamingResponse:
+    """Same as /events, as Server-Sent Events: `status` (agent steps), `delta` (answer as it is written),
+    `reset`, then `done` with the final replies (identical to /events). `error` if the engine failed."""
+    await _require_patient(token)
+    queue: asyncio.Queue[tuple[str, Any]] = asyncio.Queue()
+    marker = agent_stream.STREAM.set(queue)  # copied into the task's context below
+    try:
+
+        async def run() -> None:
+            try:
+                replies = await get_engine().handle(token, body)
+                queue.put_nowait(("done", [r.model_dump() for r in replies]))
+            except Exception:  # noqa: BLE001 — reported to the client, which falls back to /events
+                log.exception("dialog stream failed")
+                queue.put_nowait(("error", None))
+
+        task = asyncio.create_task(run())
+    finally:
+        agent_stream.STREAM.reset(marker)
+
+    async def events():
+        try:
+            while True:
+                try:
+                    kind, payload = await asyncio.wait_for(queue.get(), timeout=15)
+                except TimeoutError:
+                    yield ": keepalive\n\n"
+                    continue
+                yield f"event: {kind}\ndata: {json.dumps(payload, ensure_ascii=False)}\n\n"
+                if kind in ("done", "error"):
+                    return
+        finally:
+            if not task.done():
+                await task
+
+    return StreamingResponse(
+        events(),
+        media_type="text/event-stream",
+        headers={"Cache-Control": "no-cache", "X-Accel-Buffering": "no"},  # nginx: do not buffer SSE
+    )
+
+
 @router.get("/dialog/{token}/outbox", response_model=DialogOut, tags=["dialog"])
 async def dialog_outbox(token: str) -> DialogOut:
     """System-initiated replies (reminders, check-ins) waiting for the web console."""