That already refers to a kind of specific girlfriend, weirdly enough. Even if you don't know her at all and she may in fact not exist. It's just somehow more "certain" than the out-of-the-blue Do you have a girlfriend?, which is more "any" girlfriend.
The same goes for those sentences where we have a gadget, I think it was a camera or a cell phone, and we're asking either whether you have one (partitive) or whether you have a Japanese one (the "whole" object, which here looks like nominative).
I know this may sound strange and very illogical, but it's perfectly logical to us... :-/