Modeling tiling problems as SAT problems
One example of a tiling problem that was successfully attacked by reducing it to a SAT instance was rectangular grid coloring.  In "Extremely Complex 4-Colored Rectangle-Free Grids:
Solution of Open Multiple-Valued Problems" the authors describe how they tackled 17x17, 17x18, and 18x18 grid colorings using a SAT solver plus some previous knowledge of grid coloring theory to pare down the search space.