  The engineering of Majorana zero modes in topological superconductors, a new
paradigm for the realization of topological quantum computing and
topology-based devices, has been hampered by the absence of materials with
sufficiently large superconducting gaps. Recent experiments, however, have
provided enthralling evidence for the existence of topological surface
superconductivity in the iron-based superconductor FeSe$_{0.45}$Te$_{0.55}$
possessing a full $s_\pm$-wave gap of a few meV. Here, we propose a mechanism
for the emergence of topological superconductivity on the surface of
FeSe$_{0.45}$Te$_{0.55}$ by demonstrating that the interplay between the
$s_\pm$-wave symmetry of the superconducting gap, recently observed surface
magnetism, and a Rashba spin-orbit interaction gives rise to several
topological superconducting phases. Moreover, the proposed mechanism explains a
series of experimentally observed hallmarks of topological superconductivity,
such as the emergence of Majorana zero modes in the center of vortex cores and
at the end of line defects, as well as of chiral Majorana edge modes along
certain types of domain walls. We also propose that the spatial distribution of
supercurrents near a domain wall is a characteristic signature measurable via a
scanning superconducting quantum interference device that can distinguish
between chiral Majorana edge modes and trivial in-gap states.
