We study a family of periodically weighted Aztec diamond dimer models near their turning points. We establish that, asymptotically, as N→∞, their fluctuations there, scaled by √N, are described by a marked GUE-corners process. This limiting point process is constructed by assigning a Bernoulli mark independently to each particle in a realization of the GUE-corners process. The Bernoulli parameters associated with the random marks reflect the periodicity of the model in the limit. To prove this result we use a double-contour integral representation of the inverse Kasteleyn matrix on a higher-genus Riemann surface, which is well-suited for asymptotic analysis.