POST requests fail when <sessionState cookieless="AutoDetect" /> is set

The only solution I can see to this is to append AspxAutoDetectCookie=1 to all POST requests.

This way, ASP.NET will never redirect the request and we can dodge the 302 vs 307 question altogether. If cookies are embedded in the request, ASP.NET will detect that cookies are supported, and if no cookies are embedded, it will assume they aren't.