There is a one-to-one correspondence between a periodic domino tiling and a ground state configuration of the fully-frustrated Ising model on a two-dimensional periodic lattice. To see that, we note that at the ground state, each plaquette of the spin model must contain exactly one frustrated interaction. Therefore, viewing from the dual lattice, each frustrated edge must be "covered" by a 1x2 rectangle, such that the rectangles span the entire lattice and do not overlap, or a domino tiling of the dual lattice.