I'm no legal expert, but both countries acknowledge the copyright exists at the moment the creation is saved to a fixed medium, right?
What I would probably do is exactly the same thing I'd do if the collaboration existed in a single country--leave copyright out of it until I had a request for the work from an agent or production company based on my query.
Before I (or my partner) sent it out, I would begin the copyright registration process in the country where the agent or company does business.
I suppose the same might apply to contest entries in which you send in the play; register your copyright, if you feel you should, it the nation where the contest is.
Maryn, who did not register a copyright on her lone play, since she trusted the theatre
Edit to add: Someone has asked me what happened with my play. Nothing bad. It was performed in a black-box setting with Equity actors in front of a paying audience, one performance. The theatre which hosted the contest is one where I've had season tickets for eons, so I felt I was safe entrusting them. I saw things in performance that I hadn't seen on the page, of course, and did a rewrite which I seem to be keeping all to myself. Such a coward, huh?