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Probably more inertia than magnetism. The water isn't stuck to the bowl as a solid mass. Rotating with the bowl requires the water to drag on itself to move with the bowl. That takes time for enough force to transfer to really get the water rotating at any significant speed, so it merely looks like it's acting like a compass. If you keep rotating, the water will eventually catch up to the bowl's speed.
Mind, anything floating on the water could also actually act like a compass needle if it has a strong enough charge differential across it. That's not uncommon at all, it'd just be weird for spit or a collection of bubbles to have enough charge since they're bad at holding charge.
The test would be to spin around once or twice with the bowl and set it down. If it is not magnetic, the water should stay rotating very slowly and the slobber/etc will just continue to rotate a little and eventually stop and sit at a new orientation. If it IS acting like a compass, the floaters will slow down and then rotate back the few degrees the other way to stay oriented.
Yes, just inertia explains what's described. I'm in the habit of drinking a hot beverage that sometimes has bits of stuff floating in it, and if there's one floating bit I want to slurp up right away the practical way to get it over to the side of the cup I'm drinking from is to turn around until I'm on the same side of the cup that it is. I rotate, the liquid doesn't. It's' true that "the water will eventually catch up" to the rotation of the bowl, but it might take quite a lot of rotations before it gains any appreciable momentum. The viscosity of water is relatively low.
Personally, I just got good at the cup swirl motion. Enough to get it rotating without the wave going over the top. Mostly just the lazier option but it is much less convenient to turn yourself when at a desk. lol