So I had this image of posts 1/4" in size maybe and on the bottom would be a tab that stuck out in one direction. Then, tiles would be made such that the corners where the posts would go would be about 3 cardstock thicknesses lower than the rest of the tile. There'd be a riser attached to the cardstock that wraps the tile that is 2 layers of cardstock thick. It would be designed so that the tab on the bottom of the posts would slip under the texture and hopefully stay in place. Then, walls could be pinned to the posts, or maybe they'd slot together kinda TLX-like.
I'm thinking you could print a template onto plain copy paper and paste that into foamcore. That would give you a gluing guide for the corner textures and where to put the riser in the middle. Then, you'd rap the whole thing in cardstock.
Rather than having foamcore tabs to connect the tiles, just use 2x thickness cardstock tabs.
I realize anyone reading this probably can't follow my babbling, but I'll be able to remember it tomorrow, and I'll be able to try it out pretty quickly. Could this be a workable solution for having NO ANCHORS, and even NO TABS? That would mean just cutting the foamcore tiles square without notches for tabs... We'll see.