Highly ionic environments can mediate unusual chemical reactions that would otherwise be considered impossible based on chemical intuition. For example, the formation of a chemical bond between two iodide anions to form a divalent polyiodide anion is seemingly prohibited due to Coulombic repulsion. Using ab initio molecular dynamics simulations, we show that in the 1-ethyl-3-methylimidazolium (EMIM)/I3 ionic crystal, the reactive formation of divalent and even trivalent polyiodide anions occurs with extremely small energetic barriers, due to the electrostatic field of the ionic lattice. A practical consequence of this anomalous reactivity is that iodide anions are efficiently transported within the crystal through a "Grotthuss-exchange" mechanism involving bond-breaking and forming events. We characterize two distinct transport pathways, involving both I42- and I73- intermediates, with fast transport of iodide resulting from the release of an I- anion on the opposite side of the intermediate species from the initial bond formation. The ordered cation arrangement in the crystal provides the necessary electrostatic screening for close approach of anions, suggesting a new counterintuitive approach to obtain high ionic conductivity. This new design principle could be used to develop better solid-state electrolytes for batteries, fuel cells, and supercapacitors.