Stargate-like Background... How?

So, I'm trying to build a game like Stargate, using Godot:

https://www.youtube.com/watch?v=N4BW5GcRW44

The screen and map repeat themselves.  So, for instance, if I move far enough in one direction or the other I will eventually arrive back where I started--from the opposite direction. I'm using a parallax background currently, which gives this illusion through mirroring.  And, the background does repeat like I want, but the other objects (like enemy units and such) never reappear unless I reverse back to them the way I came.

I've seen plenty of tutorial on how …