When I run "mate" to open the project in the first display TextMate now correctly sets focus on the project in the first display. But it also shows the project in the second display, covering the Terminal.

Is it possible to not show the project on the second display?

Not sure this is remotely relevant, but it might be worth changing 'Displays have separate spaces' in Mission Control.

I have a colleague who was seeing some strange behaviour related to focusing, using multiple displays

Unticking this option fixed his problems

Admittedly it's a bit of a long shot, but worth a try at least