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Near-threshold production of antihydrogen ion ( ${\bar{\mathrm{H}}}^{+}$ ) in positronium–antihydrogen collisions is predicted by a rigorous four-body scattering calculation. The convergence of the cross sections for the rearrangement and for all competing reactions (elastic scattering, de/excitation and de/polarization of positronium) is carefully examined against partial waves. The multi-channel scattering solutions are composed of functions that diagonalize the four-body Hamiltonian, and scattering functions that satisfy the correct asymptotic boundary conditions. The production rates of ${\bar{\mathrm{H}}}^{+}$ show large discrepancies compared to more approximate calculations.