Ok, I got something that works! It's not pretty, but it definitely does the job.
Create a QMainWindow derivative with all of your actual GUI code in it and overload the event() function of this class to call this->show()
Create a class (let's call it Runner) which will hold a pointer to your QMainWindow derivative, and give it a run function.
In the Runner::Runner(), start up a thread which will call Runner::run()
In Runner::run() (which is now running in it's own thread) construct a QApplication, and an instantiation of your QMainWindow derivative. Call the exec() function of the QApplication.
Now, when you want to start up your GUI, just post any event to your QMainWindow derivative, and it will show itself!
This solution seems to work very well in Linux, although it really seems to be exploiting some loophole in Qt and may not work on other platforms. Definitely easier than patching Qt though.