The liquid structure and electrical screening ability of ionic liquids are fundamentally intertwined. The molecular nature of the charge carriers means that screening distances of external fields depend sensitively on the ion packing and structure of the ionic liquid. In this work, we quantitatively illustrate how the liquid structure itself is directly modulated by electrostatic screening conditions. In particular, electronic polarization fundamentally relaxes long-range ion structuring in asymmetric ionic liquids such as [BMIM+][BF4-], with the influence propagating to short-range ion-ion correlation. A consequence of the exact Stillinger-Lovett second moment condition is that, at fixed density, any pairwise-additive, nonpolarizable force field will necessarily predict artificially enhanced long-range ion structuring. This is because the screening condition is set by the infinite-frequency dielectric response. There is no ad-hoc fix: One has to use polarizable force fields to correctly reproduce the optical dielectric constant. Our illustration of this fundamental effect significantly clarifies interpretation of previous work comparing property prediction using polarizable and nonpolarizable force fields.