  We demonstrate that recent angle-resolved photo-emission spectroscopy
experiments provide strong evidence for the existence of two competing
topological superconducting phases in FeSe$_{0.45}$Te$_{0.55}$. The coupling of
their underlying microscopic mechanisms -- one based on a three-dimensional
topological insulator, one based on two-dimensional superconductivity --
stabilizes topological superconductivity over a wide range of parameters, and
gives rise to two disjoint topological regions in the phase diagram of
FeSe$_{0.45}$Te$_{0.55}$. We show that the topological origin of these regions
can be identified by considering the form of Majorana edge modes at domain
walls.
