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-cli/webber_cli/client.py

@@ -3,7 +3,9 @@ Webber API client.
 
 Communicates with the Webber API backend for agent execution.
 """
+import json
 import httpx
+from collections.abc import AsyncIterator
 from dataclasses import dataclass
 from typing import Any
 
@@ -132,6 +134,52 @@ class WebberClient:
             error=data.get("error"),
         )
 
+    async def run_agent_stream(
+        self,
+        agent_type: str,
+        prompt: str,
+        working_dir: str = ".",
+    ) -> AsyncIterator[str]:
+        """
+        Run an agent with streaming response.
+
+        Args:
+            agent_type: Type of agent (e.g., "explore")
+            prompt: User prompt/query
+            working_dir: Working directory for the agent
+
+        Yields:
+            Text chunks as they arrive
+        """
+        # Use a fresh client for streaming with longer timeout
+        async with httpx.AsyncClient(
+            base_url=self.base_url,
+            timeout=httpx.Timeout(300.0, connect=10.0),
+        ) as client:
+            async with client.stream(
+                "POST",
+                "/agents/stream",
+                json={
+                    "agent_type": agent_type,
+                    "prompt": prompt,
+                    "working_dir": working_dir,
+                },
+            ) as response:
+                response.raise_for_status()
+                async for line in response.aiter_lines():
+                    if line.startswith("data: "):
+                        try:
+                            data = json.loads(line[6:])
+                            event = data.get("event")
+                            if event == "chunk":
+                                yield data.get("data", "")
+                            elif event == "error":
+                                raise Exception(data.get("data", "Unknown error"))
+                            elif event == "done":
+                                break
+                        except json.JSONDecodeError:
+                            continue
+
     async def __aenter__(self) -> "WebberClient":
         """Async context manager entry."""
         return self