If an animal tiles the plane via translation, can it do so in a lattice configuration?

Question

It is known that if a polyomino tiles the plane using only translated copies, then it has at least one such tiling where the centroids of each tile form a lattice; see for instance the paper Arbitrary versus periodic storage schemes and tessellations of the plane using one type of polyomino. (In fact, this result extends to topological disks with piecewise $C^2$ boundary and a finite number of inflection points; see Tiling the plane with one tile, by D. Girault-Beauquier and M. Nivat.)

I am curious whether this result holds if we relax the connectivity requirement on polyominoes, and consider arbitrary animals in the plane, i.e. finite subsets of $\mathbb{Z}^2$.

Are there any animals which tile the plane via translation, but only in an anisohedral manner?

Answer 1

In fact the answer to this question turns out to be trivial. Consider the simplest possible disconnected animal: two squares separated by a distance of 2.

 _____        _____
|     |      |     |
|     |      |     |
|_____|      |_____|

This animal tiles the plane by translation, but any such tiling must within a horizontal line have tiles whose $x$-coordinates go $n, n+1, n+4,n+5,n+8,n+9,\ldots$, and so cannot form a lattice.

However, if we relax to periodic tilings, the answer to the title question turns out to be yes - every finite subset of $\mathbb{Z}^2$ that admits a tiling does so in a periodic manner, as proven in Bhattacharya 2016. (Credit to this post of Terrence Tao's for alerting me to the existence of this paper.)