feat: add streaming responses to API and CLI

Add real-time streaming support for agent responses using Server-Sent
Events (SSE). Responses now appear as they're generated instead of
waiting for completion.

- Add run_stream method to BaseAgent and ExploreAgentImpl
- Add /agents/stream SSE endpoint to API router
- Add run_agent_stream method to CLI client
- Add --stream flag to chat and explore commands (enabled by default)
- Use --no-stream for batch mode with spinner

Co-Authored-By: Claude Opus 4.5 <noreply@anthropic.com>

webber-api/src/domains/agents/router.py

@@ -1,7 +1,9 @@
 """
 REST API routes for agents.
 """
+import json
 from fastapi import APIRouter, HTTPException
+from fastapi.responses import StreamingResponse
 
 from src.domains.agents.base import get_agent, list_agents
 
@@ -68,6 +70,53 @@ async def run_agent(request: AgentRunRequest) -> AgentRunResponse:
         )
 
 
+@router.post("/stream")
+@logged()
+async def stream_agent(request: AgentRunRequest) -> StreamingResponse:
+    """
+    Run an agent with streaming response.
+
+    Returns Server-Sent Events (SSE) with text chunks.
+    Event types:
+    - "chunk": Text chunk from the agent
+    - "done": Stream complete
+    - "error": Error occurred
+    """
+    agent = get_agent(request.agent_type)
+    if not agent:
+        raise HTTPException(
+            status_code=400,
+            detail=f"Unknown agent type: {request.agent_type}"
+        )
+
+    async def generate():
+        try:
+            async for chunk in agent.run_stream(
+                request.prompt,
+                working_dir=request.working_dir,
+            ):
+                # SSE format: data: {json}\n\n
+                event = {"event": "chunk", "data": chunk}
+                yield f"data: {json.dumps(event)}\n\n"
+
+            # Signal completion
+            yield f"data: {json.dumps({'event': 'done'})}\n\n"
+
+        except Exception as e:
+            logger.exception(f"Stream error: {e}")
+            error_event = {"event": "error", "data": str(e)}
+            yield f"data: {json.dumps(error_event)}\n\n"
+
+    return StreamingResponse(
+        generate(),
+        media_type="text/event-stream",
+        headers={
+            "Cache-Control": "no-cache",
+            "Connection": "keep-alive",
+        }
+    )
+
+
 @router.get("/{agent_type}", response_model=AgentInfo)
 async def get_agent_info(agent_type: str) -> AgentInfo:
     """Get information about a specific agent."""