Non-overlapping tilings of regions is a well-studied topic. I wonder if the following variant has been considered:

A tile can be partitioned into several regions, where such regions from different tiles may overlap, according to some set of rules, similar (but not the same) as pieces of a puzzle.

A natural example would be to consider all areas covered by industrial machine as a tile, but not all parts of the machine are active at the same time. Thus, not all regions around the machine are occupied at the same time, which allows one to put several such machines closer together, as long as two "active" regions do not overlap.

From a mathematical point if view, the following is a natural question: Given a finite set of rectangular tiles, each with rectangular regions, and a set of rules that determine how the regions may overlap, together with some larger region of the plane, what is the complexity to determine if the tiles can fit inside this large region without violating the rules?

This sounds like an NP-complete problem.