Positron binding energies in halogenated hydrocarbons are calculated ab initio using many-body theory, elucidating the role of the anisotropic positron-molecule potential and the comparative effect of fluorination vs. chlorination. For chlorinated molecules, including planar molecules for which the interaction is highly anisotropic, excellent agreement with experiment and recent DFT-based model-potential calculations of [Suzuki et al., Phys. Rev. A 102, 052830 (2020)] is found. It is shown that their fluorinated counterparts generate a less attractive positron-molecule potential due to higher molecular orbital ionization energies and smaller density of occupied valence molecular orbitals, resulting in very weak, or in most cases loss of, positron binding.