### What happened?

`MidStreamFallbackError` raised mid-stream during `Router.aresponses(stream=True)` bypasses the Router's fallback chain. Configured cross-provider fallbacks (e.g. `anthropic → vertex_ai`) never fire when the primary provider's stream fails mid-flight.

Observed in production: an Anthropic socket timed out before the first chunk on `Router.aresponses(stream=True)`. The underlying `CustomStreamWrapper` raised `MidStreamFallbackError` from [`_handle_stream_fallback_error`](https://github.com/BerriAI/litellm/blob/main/litellm/litellm_core_utils/streaming_handler.py) — intended behavior, that's exactly what's supposed to drive fallback. The Router's configured `anthropic → vertex_ai` fallback was not invoked; the error surfaced unhandled to the caller.

### Root cause

The chat completions path wraps its `CustomStreamWrapper` via [`Router._acompletion_streaming_iterator`](https://github.com/BerriAI/litellm/blob/main/litellm/router.py), which catches `MidStreamFallbackError` and re-enters `async_function_with_fallbacks_common_utils`.

The Responses API path doesn't have an equivalent: - `Router.aresponses` is bound via `factory_function(litellm.aresponses, call_type="…