How can I reverse proxy a subdomain to the root of another host without rewriting all the HTML?

I've set up nginx to proxy http://example.com/subdomain to http://myrealserver.com. It works as far as contacting myrealserver but the responses have myrealserver.com or the IP address or an encoded and gzipped URL, or... you get the picture.

I have no control over the myrealserver.com web app as it's built into an embedded device. I've spent a while trying the various nginx rewrite options but the URL is hidden in so many ways that it's proving very difficult to track them all down.

Is there a better way to do it?