It seems to my intuition like there should only be one “tidal bulge”, directly under the Moon. But there's another one on the far side. Why?
Earth and the Moon are both in orbit around the center of gravity of the Earth-Moon system. Earth is in free fall toward the Moon. As the Moon turns around the Earth, the entire Earth is pulled toward the Moon. The average gravitational pull of the Moon is accounted for by this displacement of the Earth.
So, the tidal force at any given point on Earth's surface is not determined by the absolute gravitational pull of the Moon, but the deviation from the average gravitational pull of the Moon at that point. The gravitational pull is stronger than average directly under the Moon, creating a bulge toward the Moon there, and weaker than average on the far side of Earth, creating a bulge away from the Moon there.
Remember, again, the Earth is in free fall toward the Moon. It isn't that the water on the far side has moved away from the Moon - the Earth has moved away from that water.
Rotation has nothing to do with it - the far side bulge is not, as is commonly held, a consequence of inertia or centrifugal force.
See https://www.av8n.com/physics/tides.htm, https://earthscience.stackexchange.com/a/16557
There are tides on land too!