The Domino problem is undecidable on every rhombus subshift

We extend the classical Domino problem to any tiling of rhombus-shaped tiles. For any subshift X of edge-to-edge rhombus tilings, such as the Penrose subshift, we prove that the associated X-Domino problem is ${\mathrm{\Pi }}_1^0$ -hard and therefore undecidable. It is ${\mathrm{\Pi }}_1^0$ -complete when the subshift X is given by a computable sequence of forbidden patterns.