You know you’re an econ/math nerd if you read this and think “haha, it’s like the matching pennies game”:
But hear me out…here’s the matching pennies game, and, like the joke, the crux of the game is that there is no Nash equilibrium without randomization. To further the analogy: The matching pennies game works as it does because player 1, let’s say, “gets off” when the pennies match whereas player 2 gets off when the pennies don’t match. (This wording hopefully shows the intuition of why there is no pure-strategy Nash equilibrium, since the goals of the players are clearly mutually exclusive.) The…uh, matching fetishes game works in a similar fashion, since the guy gets off when he and his partner are in agreement, but the woman gets off when there is discord.
With some minor labeling changes, you could even make a payoff matrix for the matching fetishes game, the conclusion of which is…hm, can one randomize being turned on or not?