The secret is in no change in the gravitational pull of the moon, but in the direction of the vector in relation to the Sun's gravity and the combined effect of both on the ground is all .
When we see the full moon is fully illuminated by the Sun , while the side we see from Earth is in darkness .
Put another way , the full moon occurs at a maximum distance of the moon opposite the sun, at the confluence with the Sun in opposing forces on Earth that lies directly between them.
This is the explanation for the rising tides , and concomitant RIP forces in the plates themselves as the Earth revolves focused monthly gravimetric field .
At full moon, the sun's and the moon's gravity act in opposite directions, putting maximum tension on the rocks in the Earth's crust. Add in where the Jovian planets are in relation to the Earth, and there may be even more tension to trigger earthquakes.
For all of you saying the correlation is total bunk, please read this: