What is the approximate maximum separation measured along the surface of the Earth between two stations communicating by EME?

Explanation

Both stations must have the Moon above the horizon at the same moment, and at any instant the Moon is visible from very nearly half the Earth, so the two stations can be almost antipodal. Perigee and apogee change the path loss rather than the geometric limit, so the answers that tie the separation to the Moon's distance are wrong.