The solution is to add the forms authentication cookie when performing a HttpWebRequest.
Uri uri = new Uri("http://services.mysite.com/document/documentservice.svc");
HttpWebRequest webRequest = (HttpWebRequest)WebRequest.Create(uri);
HttpCookie cookie = HttpContext.Current.Request.Cookies[FormsAuthentication.FormsCookieName];
Cookie authenticationCookie = new Cookie(FormsAuthentication.FormsCookieName, cookie.Value, cookie.Path, HttpContext.Current.Request.Url.Authority);
webRequest.CookieContainer = new CookieContainer();
webRequest.CookieContainer.Add(authenticationCookie);
WebResponse myResponse = webRequest.GetResponse();
Stream stream = myResponse.GetResponseStream();