Suppose you have two Euclidean planes, and there is some set of pairs of points (themselves pairs of coordinates) such that for each such pair ((ax,ay),(bx,by)), the line segment connecting the point (ax,ay) to (bx,by) on plane 1 is identified with the line segment connecting those points on plane 2.
Then if that shared line segment is the edge of some tile (closed polygon) on one plane, and of another tile on the other plane, those tiles are said to be adjacent, just as two tiles sharing an edge on the same plane are adjacent. Further, we shall require that every such shared line segment is an edge of some tile.
Say then that the pair of planes together with the set of all tiles and all pairs of identified edges is a "2-planar tiling" if: 1. every edge is shared by exactly two tiles; 2. no two tiles on the same plane ever overlap; and 3. any tile on either plane can be reached from any other tile on either plane by some sequence of steps between adjacent tiles.
My question is: is there a 2-planar tiling, by this definition, which is topologically distinct from any possible tiling on just one plane? That is, having some subset of tiles which cannot occur on a single plane with the same pattern of adjacencies? If there is, what is the "most regular" such tiling?