OK, my glove box door hinge broke on the right side, and I see from googling all the audi forums that this is not an uncommon failure.
I have devised a fix that doesn't require removing the whole glove box. Some of you may not like the inelegance of the solution, so those of you shouldn't do it.
As always, no guarantees, you're on your own if your car explodes or you die.
First, snap off the side panel as instructed here:
I also removed the strut and disassembled it as explained in that post. I put some light grease inside to make it less stiff.
The first photo shows my broken glove box.
The second photo shows the screw I bought at the hardware store, a self-tapping screw made for concrete use. A pack of 3 was about $2.
The third photo shows where to drill a hole. You drill a 3/16 diameter hole through the glove box door and all the way through the hinge. The hole is about 5/8 to 3/4 from the bottom edge of the door. The hole wants to line up side-to-side with the left slot in the hinge (shown with the yellow arrow).
The fourth photo shows the drilling process. You need to reach in the side and hold the hinge in position so it doesn't flip back while you are drilling. Extend the drill bit as far as you can in your drill, you need to drill about 3" deep.
The fifth photo shows a trial installation. The screw cuts thread in the plastic.