This feels like a niche market someone needs to go after. I would love to have a bathroom floor or a kitchen backsplash tiled in 'hat' in 4 colours (ensuring no two same-colour tiles touched of course).
I suspect, given the 1:1 correspondence to a hex grid (where some of the hexagons map to an inverted/non inverted pair of hats) that they describe in the original paper that it would be possible to tile with just three colors of noninverted hat, and one color of inverted hat.