After a couple of months of work, we cracked it: If you only allow yourself to tile by translations and rotations, then Tile(1,1) admits only non-periodic tilings! We call this a "weakly chiral aperiodic monotile" -- it's aperiodic in a reflection-free universe, but tiles periodically if you're allowed to use reflections.
The tiling is reminiscent of, but not identical to, hat tilings -- it contains a sparse population of "odd" tiles, which are rotated by odd multiples of 30 degrees relative to all other tiles. (5/n)