I would play for both teams and hope you aren't drawn against yourself or play for the team with fewer players so as to have the maximum number of games
Member of two teams

I am not sure how that works? You might not be able to participate at all? I remember this coming up with another chess player a while back but I guess you will find out.
As far as ethically picking one presuming you can? Me personally I would go option 3 and play for my favourite.

I was thinking of suggesting a match with the Picts to my BigFooty club - it would make a good match. Anyway it wouldn't be for a while yet.

Not a bad idea Pickler, I have my mental health club which I have contemplated challenging the Picts, we just need a few more big hitters and we will be ready.
Maybe we could have a game in the lead up to it? We are another few weeks off being ready but we could have a decider match, then the winner plays the Picts with Davies approval?

The Bigfooty team only plays one game at once and there is one on now with still a long way to go. My opponent timed out right at the start which was disappointing. I could join your club if you like, if I would qualify. Luckily I don't have mental health issues.

That is a good idea to only play one at a time, prevents clogging everyone up.
It would be great if you wanted to join, don't need to have issues, the fact that you are showing your support is enough. Plus we are reasonably active with chess activity.
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Just a theoretical question at the moment, but what should one do if one is a member of both teams playing a match? I don't imagine there are any rules, so it's just a question of etiquette.
1) sit it out
2) play for the team with fewer players so as to have the maximum number of games
3) just play for who you like
4) play for both teams and hope you aren't drawn against yourself