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Iron-based binary chalcogenide superconductors FeSe and FeS have attracted much recent attention due to their complex orbital-selective correlations and Cooper pairing, offering the minimal model system holding the key properties to understanding the physics of high-Tc superconductors. Here, using density functional plus dynamical mean-field theory method with full self-consistency over the charge density, we study the effect of electronic correlations on the electronic structure, magnetic properties, orbital-dependent band renormalizations, and Fermi surface of the tetragonal phase of bulk FeS. We perform a direct structural optimization of the P4/nmm crystal structure of paramagnetic FeS, minimizing the total energy of FeS with respect to the lattice constant a and the internal coordinate zS of atom S. Our results show an anomalous sensitivity of the electronic structure and magnetic properties of FeS (intrinsic to all Fe-based chalcogenide and pnictide superconductors) to fine details of its crystal structure, e.g., to a small variation of the chalcogen coordinate zS. Upon expansion of the lattice volume (which can be realized, e.g., in FeS1-xSex and FeS1-xTex), we observe a remarkable change of the electronic structure of FeS which is associated with a complete reconstruction of the Fermi-surface topology (Lifshitz transition). This behavior is ascribed to a correlation-induced shift of the Van Hove singularity associated with the Fe xy and xz/yz orbitals at the M point across the Fermi level. The Lifshitz phase transition is accompanied by a significant growth of local magnetic moments and emergence of strong orbital-selective correlations. It is seen as a pronounced anomaly ("kink") in the total energies upon expansion of the lattice, associated with a remarkable enhancement of compressibility. This behavior is accompanied by an orbital-dependent formation of local moments, a crossover from itinerant to localized orbital-selective moment behavior of the Fe 3d electrons. While exhibiting weak effective mass enhancement of the Fe 3d states m∗/m∼1.3-1.4, correlation effects reveal a strong impact on a position of the Van Hove singularity originating from the Fe xz/yz and xy orbitals at the M point, implying a complex interplay between electronic correlations and band structure effects in FeS. Our results suggest a complex interplay between electronic correlations, magnetism, and lattice degrees of freedom in FeS. © 2019 American Physical Society.