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Determinantal point processes are well-known for describing multipoint behavior in \beta=2 random matrix models. The sine process is the universal determinantal process appearing in the bulk of the spectrum of random matrices. There is a discrete version of the sine process depending on one parameter, and also its extension to two dimensions. The two-dimensional process is a unique (Sheffield 2003) translation invariant ergodic Gibbs measure on point configurations in Z^2 of a given slope. We introduce a generalization of the two-dimensional discrete sine process which in addition depends on four bi-infinite sequences of real parameters associated with the horizontal and the vertical coordinate directions (two sequences per each direction). We obtain this generalization as a bulk limit of a certain random domino tiling model coming from the fully inhomogeneous free fermion six vertex model. Based on the joint project https://arxiv.org/abs/2109.06718 with A. Aggarwal, A. Borodin, and M. Wheeler