actually, yeah. like have a sliding, folding door in front and fake windows on the side, so it will actually look like a phone booth when the door is closed. then when you open the door, you can step in and play on the arcade machine inside the booth.
Hmmm... Interesting idea. If you actually "stepped in", the TARDIS would have to be pretty big (about as big as the actual TARDIS prop), and it would be a little claustrophobic, especially with two players. Also, other people couldn't really watch from the sides. And I just don't have that kind of room.
As a variation on that, I suppose the doors could swing open to reveal the screen directly behind them (but still angled back like a typical arcade cab), and the motion of the doors opening could extend the CP out toward you on a sliding track (and the CP would retract back in as you close the doors). I'd still have the problem of not being able to see the screen from the sides, since the side walls wouldn't be cut away.
This is probably getting a little complex, with some usability problems. But thanks for the idea.