  Fission fragments' charge and mass distribution is an important input to
applications ranging from basic science to energy production or nuclear
non-proliferation. In simulations of nucleosynthesis or calculations of
superheavy elements, these quantities must be computed from models, as they are
needed in nuclei where no experimental information is available. Until now,
standard techniques to estimate these distributions were not capable of
accounting for fine-structure effects, such as the odd-even staggering of the
charge distributions. In this work, we combine a fully-microscopic collective
model of fission dynamics with a recent extension of the particle number
projection formalism to provide the highest-fidelity prediction of the primary
fission fragment distributions for the neutron-induced fission of $^{235}$U and
$^{239}$Pu. We show that particle number projection is an essential ingredient
to reproduce odd-even staggering in the charge yields and benchmark the
performance of various empirical probability laws that could simulate its
effect. This new approach also enables for the first time the realistic
determination of two-dimensional isotopic yields within nuclear density
functional theory.
