After core ionization of an atom or molecule by an x-ray photon, multiply charged ions are produced in the Auger decay process. These ions tend to neutralize their charge when embedded in an environment. We demonstrate that, depending on the atom or molecule and its neighbors, electron transfer mediated decay (ETMD) provides a particularly efficient neutralization pathway for the majority of the ions produced by Auger decay. The mechanism is rather general. As a showcase example, we conducted an ab initio study of the NeKr2 cluster after core ionization of the Ne atom. This example has been chosen because it is amenable to both ab initio calculations and coincidence experiments. We find that even for frozen nuclei, the neutralization rate can be as fast as 0.130  ps(-1). We also show that nuclear dynamics may increase the rate by about an order of magnitude. The generality of the mechanism makes this neutralization pathway important in weakly bonded environments.