Examples of concrete games to apply Borel determinacy to

I'm teaching a course on various mathematical aspects of games, and I'd like to find some examples to illustrate Borel determinacy. Open or closed determinacy is easy to motivate because it proves the determinacy of a game like chess, idealized to make infinitely long games a draw. But I have a hard time trying to come up with examples of not-too-theoretical games to which I could apply Borel determinacy.

Specifically, I'm looking for examples of:

1. games that can be cast as Gale-Stewart games,

2. which don't look too much like an abstract theoretical construct but like an actual concrete game that people might play, at least granting them an infinite amount of time (but each step of the game should be more or less finitistic in nature),

3. whose determinacy follows from the Borel determinacy theorem,

4. but not from the open or closed determinacy theorem alone,

5. and whose winning strategy is not obvious by other means.

If we drop condition (4), then the idealized game of chess is a good example. If we drop condition (5), an example might be the game where Alice and Bob play binary digits and Alice wins if the proportion of 1's has a limit (this stretches condition (2) a bit but I'm willing to allow it… my problem is mostly that Alice's winning strategy is really too obvious).

Can anyone offer interesting examples with all these conditions?