Transport of coherent waves in multiple-scattering media may exhibit fundamental, nonintuitive phenomena such as halt of diffusion by disorder called Anderson localization. For electromagnetic waves, this phenomenon was observed only in one and two dimensions so far. However, none of these experiments studied the contribution of reciprocal paths nor their manipulation by external fields. In order to weaken the effect of reciprocity of coherent wave transport on Anderson localization in one dimension, we studied light propagation through stacks of parallel Faraday-active glass slides exposed to magnetic fields up to 18 T. Measurements of light transmission statistics are presented and compared to one-dimensional (1D) transfer-matrix simulations. The latter reveals a self-organization of the polarization states in this system leading to a saturation of the Faraday rotation-induced reciprocity breaking, an increase of the localization length, and a decrease of transmission fluctuations when reciprocity is broken. This is confirmed experimentally for samples containing small numbers of slides while for larger samples a crossover from a 1D to a quasi-1D transport regime is found.