Domino tilings of black-and-white Temperleyan cylinders

We consider the dimer model in cylindrical domains $\Omega_\delta$ on square grids of mesh size $\delta$ with two Temperleyan boundary components of different colors. Assuming that the $\Omega_\delta$ approximate a cylindrical domain $\Omega$ as $\delta\to 0$, we prove the convergence of height fluctuations to the Gaussian Free Field in $\Omega$ plus an independent discrete Gaussian multiple of the harmonic measure of one of the boundary components. The limit of the dimer coupling functions on $\Omega_\delta$ is holomorphic in $\Omega$ but not conformally covariant. Given this, we determine the limiting structure of height fluctuations from general principles rather than from explicit computations. In particular, our analysis justifies the inevitable appearance of the discrete Gaussian distribution in the doubly connected setup.