The good news is, invisible models will be duplicated too. I'd suggest putting some invisible trigger models near the edge of the terrain. If the player passes these invisible models, you know he is onto the next "square" of the terrain, and so remove or set INVISIBLE|PASSABLE those objects you don't want carried over.

You'd have to do it cleverly because I'd imagine you want him to be able to return to the original "square" and see all the objects there just as they were. So maybe you can develop a makeshift grid system... lemme whip something up in MSPaint, hold on....