fix(anthropic): Fix streaming deadlock in Cloudflare Workers

[nanobanana123]

Summary

This PR fixes a streaming deadlock issue when using @ai-sdk/anthropic with streamText() in Cloudflare Workers.

Problem

The doStream() method uses a tee() + async first-chunk read pattern to trigger the transform stream:

const [streamForFirstChunk, streamForConsumer] = transformedStream.tee();
stream = streamForConsumer;
const reader = streamForFirstChunk.getReader();
(async () => {
  const { done } = await reader.read();
  // ...
})();
return returnPromise.promise;

This causes a deadlock in Cloudflare Workers because:

1. The stream doesn't flow until doStream() returns
2. But doStream() returns a promise that waits for the transform to process a chunk
3. The transform won't process a chunk until the stream flows
4. Deadlock

This works in Node.js but fails in CF Workers due to differences in how pull-based streams are handled.

Solution

Remove the tee() pattern and resolve the promise immediately with the stream:

stream = transformedStream;
returnPromise.resolve({
  stream,
  request: { body },
  response: { headers: responseHeaders },
});
return returnPromise.promise;

Error handling for HTTP 200 responses with error payloads is still done in the transform function via the isFirstChunk check.

Test Plan

• Tested with streamText() in Cloudflare Workers
• Verified streaming works with direct Anthropic API
• Verified streaming works through proxy configurations

Related

This is a simpler alternative to #10726 that focuses only on the minimal fix needed.

[lgrammel]

Closed in favor of #10739