How do I stream a file upload from a client to an external server with Cloudflare Workers?

I am proxying a web API that needs auth behind a Cloudflare Worker so some functions can be accessed anonymously. I have got simple POST messages working fine, but when it comes to POSTing file uploads it is unclear to me how to stream the file data from the client to the remote server.

It certainly appears that this feature is supported with Streams, per this example:

export default {
  async fetch(request, env, ctx) {
    // Fetch from origin server.
    let response = await fetch(request);

    // Create an identity TransformStream (a.k.a. a pipe).
    // The readable side will become our new response body.
    let { readable, writable } = new TransformStream();

    // Start pumping the body. NOTE: No await!

    // ... and deliver our Response while that’s running.
    return new Response(readable, response);

But I can’t really make sense of how to have this example take the incoming file upload POST and pipe it to