I am looking for two convex polygons $P,Q \subset \Bbb R^2$ such that $P$ does not tile the plane, $Q$ does not tile the plane, but if we allowed to use $P,Q$ together, then we can tile the plane.
Here I do not require the tilings to be lattice tilings, or even periodic tilings. I allow tilings by congruent copies of $P$ and/or of $Q$, i.e. I am allowing rotations and reflections!
I haven't found any example, and maybe there could be none.