We provide ab initio and density functional theory evidence for a family of surprisingly robust like-charged clusters of common HSO4- and H2PO4- oxyanions, ranging up to tetramers of net charge 4-. Our results support other recent theoretical and experimental evidence for "antielectrostatic" hydrogen-bonded (AEHB) species that challenge conventional electrostatic conceptions and force-field modeling of closed-shell ion interactions. We provide structural and energetic descriptors of the predicted kinetic well-depths (in the range 3-10 kcal/mol) and barrier widths (in the range 2-4 angstrom) for simple AEHB dimers, including evidence of extremely strong hydrogen bonding in the fluoride-bisulfate dianion. For more complex polyanionic species, we employ natural bond-orbital-based descriptors to characterize the electronic features of the cooperative hydrogen-bonding network that are able to successfully defy Coulomb explosion. The computational results suggest a variety of kinetically stable AEHB species that may be suitable for experimental detection as long-lived gas-phase species or structural units of condensed phases, despite the imposing electrostatic barriers that oppose their formation under ambient conditions.