  We use path integral molecular dynamics simulations and theory to elucidate
the interactions between charge carriers, as mediated by a lead halide
perovskite lattice. We find that the charge-lattice coupling of MAPbI$_3$
results in a repulsive interaction between electrons and holes at intermediate
distances. The effective interaction is understood using a Gaussian field
theory, whereby the underlying soft, polar lattice contributes a nonlocal
screening between quasiparticles. Path integral calculations of this nonlocal
screening model are used to rationalize the small exciton binding energy and
low radiative recombination rate observed experimentally and are compared to
traditional Wannier-Mott and Fr\"ohlich models, which fail to do so. These
results clarify the origin of the high power conversion efficiencies in lead
halide perovskites. Emergent repulsive electron-hole interactions provide a
design principle for optimizing soft, polar semiconductors.
