It’s a bit counter intuitive, but here it goes: The moon pulls on the water on Earth. But it also pulls on the Earth itself.
And that pull is a bit stronger (because it’s closer and the earth is rigid compared to water) on the Earth than on the water on the far side of the Earth. So basically the Earth’s surface “falls away” from the water on the far side because this water is subjected to the least of the moon’s gravity of all the elements in this system.
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