When fetching, the User-Agent "Mozilla/5.0 (Macintosh; Intel Mac OS X 10_9) AppleWebKit/537.71 (KHTML, like Gecko)" is used, where the browser window itself uses "Mozilla/5.0 (Macintosh; Intel Mac OS X 10_9) AppleWebKit/537.71 (KHTML, like Gecko) Version/7.0 Safari/537.71"
In web applications with (very) strict security settings, a different User-Agent string can result in a logout. I've had this happening in a CakePHP application where User-Agent checking is a default security measure.
Bug 112071 would fix this.
This is fixed now that bug 112071 is fixed. Give it a try.
I can confirm this issue fixed in Webkit Nightly r165723 "Version 7.0.2 (9537.74.9, 538+)"