We study site percolation models on planar lattices including the [m,4,n,4] lattice and the square tilings on the Euclidean plane () or the hyperbolic plane (), satisfying certain local constraints on degree‐4 faces. These models are closely related to Ising models and XOR Ising models (product of two i.i.d Ising models) on regular tilings of or . In particular, we obtain a description of the numbers of infinite “+” and “−” clusters of the ferromagnetic Ising model on a vertex‐transitive triangular tiling of for different boundary conditions and coupling constants. Our results show the possibility that such an Ising configuration has infinitely many infinite “+” and “−” clusters, while its random cluster representation has no infinite open clusters. Percolation properties of corresponding XOR Ising models are also discussed.