  This paper concludes the presentation of the non-relativistic effective field
theory formalism designed to calculate the radiative corrections that enhance
the pair-annihilation cross sections of slowly moving neutralinos and charginos
within the general minimal supersymmetric standard model (MSSM). While papers I
and II focused on the computation of the tree-level annihilation rates that
feed into the short-distance part, here we describe in detail the method to
obtain the Sommerfeld factors that contain the enhanced long-distance
corrections. This includes the computation of the potential interactions in the
MSSM, which are provided in compact analytic form, and a novel solution of the
multi-state Schr\"odinger equation that is free from the numerical
instabilities generated by large mass splittings between the scattering states.
Our results allow for a precise computation of the MSSM neutralino dark matter
relic abundance and pair-annihilation rates in the present Universe, when
Sommerfeld enhancements are important.
