Bounded lattice tiles that pack with another lattice

Suppose $L, M$ are full-rank lattices in Euclidean space, such that ${\mathrm{vol}} L < {\mathrm{vol}} M$. Answering a question of Han and Wang from 2001, we show how to construct a bounded measurable set $F$ (we can even take $F$ to be a finite union of polytopes) such that $F+L$ is a tiling and $F