I'm afraid the debug mode you are referring to is simply not AJAX.
Or you'll have to detail a bit more, what debug mode in what browser you are referring to.
As far as I can tell your question kind of says "Why can't I phone to a mobile phone on my walkie-talkie, I can get some radio stations out of it though".
It's kind of messing up a few different technologies which are "closely looking", but are actually not compatible in nature.
The manual article deals about what you can do with AJAX and how to use it.
And cross-domain prevention is still a security measure that is integrated for good reasons.