Consider a variant of Go 囲碁 which, although one player is permanently called the Black player and the other is permanently called the White player, on any turn, a player may play a stone of either color. Essentially you can kill your opponent's liberties.
I have no idea how to end the game, though it is scored normally with area scoring with each player's color.
Another variant has that, if you play an opponent's color, you pay the price of losing your next turn.
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