Interesting, I searched for that, and I ended on Tidal Acceleration
But it appears that your description is only the case for moons that orbit faster than their host planet rotates. Like Phobos moving much faster than its host, it slowly spirals down towards Mars. In turn the planet rotation speeds up (tides also exist on planets without liquid, the effect is just smaller).

The moon is slower, causes the tides on earth to move ahead of the moon which accelerates the moon and decelerates earth's rotation.