Guiding‐center kinetic theory has been developed for antihydrogen recombination experiments, which are conducted with magnetic fields of 3 – 5 T; temperatures of 4 – 10 K; positron densities of 107– 108cm−3; and antiproton densities of 104– 2 × 107cm−3. Collision operators provide the leading‐order correction to weak‐coupling theory as the coupling parameter increases. Six collision operators — three Landau analogs and three Balescu‐Guernsey‐Lenard analogs — are found for particles of unlike charges. One operator is the multiple‐species generalization of Dubin’s and O’Neil’s operator. A stability analysis is performed for counter‐streaming positrons and antiprotons occupying a cylindrical region coaxial with an outer conducting cylinder in a constant, axial magnetic field. The finite transverse geometry of the system is included, leading to to a three‐dimensional Penrose criterion, which is applied to drifting Maxwellian distributions to obtain the regime of stability as a function of the species’ temperature ratio, density ratio and relative mean velocity. Collisional corrections are considered. Terms resulting from collisions between particles of the same species cancel under general assumptions satisfied by both O’Neil’s operator and Dubin’s and O’Neil’s operator. The multiple‐species generalization of Dubin’s and O’Neil’s operator is used for unlike‐species collisions to find a collisionally corrected dispersion relation, which is applied to a detailed study of stability properties. © 2003 American Institute of Physics