  In active matter, the lack of momentum conservation makes non-reciprocal
interactions the rule rather than the exception. They lead to a rich set of
emerging behaviors that are hard to account for and to predict starting from
the microscopic scale, due to the absence of a generic theoretical framework
out of equilibrium. Here we consider bacterial mixtures that interact via
mediated, non-reciprocal interactions like quorum-sensing and chemotaxis. By
explicity relating microscopic and macroscopic dynamics, we show that
non-reciprocity may fade as coarse-graining proceeds, leading to large-scale
bona fide equilibrium descriptions. In turns, this allows us to account
quantitatively, and without fitting parameters, for the rich behaviors observed
in microscopic simulations including phase separation, demixing or multi-phase
coexistence. We also derive the condition under which non-reciprocity is strong
enough to survive coarse-graining, leading to a wealth of dynamical patterns.
Again, the explicit coarse-graining of the dynamics allows us to predict the
phase diagram of the system starting from its microscopic description. All in
all, our work demonstrates that the fate of non-reciprocity across scales is a
subtle and important question.
