The formation of positive ions of antihydrogen $bar{rm{H}}^+$ via the three body reaction (i) $rm{e}^+ + rm{e}^- + bar{rm{H}} rightarrow rm{e}^- + bar{rm{H}}^+$ is considered. In reaction (i), free positrons $rm{e}^+$ are incident on antihydrogen $bar{rm{H}}$ embedded in a gas of low-energy ($sim $ meV) electrons and, due to the positron-electron interaction, a positron is attached to $bar{rm{H}}$ whereas an electron carries away the energy excess. We compare reaction (i) with two radiative attachment mechanisms. One of them is (ii) spontaneous radiative attachment, in which the ion is formed due to spontaneous emission of a photon by a positron incident on $bar{rm{H}}$. The other is (iii) two-center dileptonic attachment which takes place in the presence of a neighboring atom B and in which an incident positron is attached to $bar{rm{H}}$ via resonant transfer of energy to B with its subsequent relaxation through spontaneous radiative decay. It is shown that reaction (i) can strongly dominate over mechanisms (ii) and (iii) for positron energies below $0.1$ eV. It is also shown that at the energies considered reaction (i) will not be influenced by annihilation and that the reaction $rm{e}^+ + rm{e}^+ + bar{rm{H}} rightarrow rm{e}^+ + bar{rm{H}}^+$ has a vanishingly small rate compared to reaction (i).
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