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Background: A quantitative microscopic understanding of the fission-fragment yield distributions represents a major challenge for nuclear theory as it involves the intricate competition between large-amplitude nuclear collective motion and single-particle nucleonic motion. Purpose: A recently proposed approach to global modeling of fission fragment distributions is extended to account for odd-even staggering in charge yields and for neutron evaporation. Method: Fission trajectories are obtained within the density functional theory framework, allowing for a microscopic determination of the most probable fission prefragment configurations. Mass and charge yields distributions are constructed by means of a statistical approach rooted in a microcanonical ensemble. Result: We show that the proposed hybrid model can reproduce experimental mass and charge fragment yields, including the odd-even staggering, for a wide range of fissioning nuclei. Experimental isotopic yields can be described within a simple neutron evaporation scheme. We also explore fission fragment distributions of exotic neutron-rich and superheavy systems, and compare our predictions with other state-of-the art global calculations. Conclusion: Our study suggests that the microscopic rearrangement of nucleons into fission fragments occurs well before the scission, and that the subsequent dynamics is mainly driven by the thermal excitations and bulk features of the nuclear binding. The proposed simple hybrid approach is well suited for large-scale calculations involving hundreds of fissioning nuclei.