Now, I'm not an artist or anything, and this was my first model. It lacks any sort of real detail or style, much less a texture. It uses 3 primitive objects.

It fits the bill. There's also a turret and barrel, but those are simple and without any real consequence.
From there, I tried to import the exported .obj file into 3do Builder+, leaving me with...

Oops.
Does anyone know what's causing this problem?