  Majorana zero modes in condensed matter systems have been the subject of much
interest in recent years. Their non-Abelian exchange statistics, making them a
unique state of matter, and their potential applications in topological quantum
computation, earned them attention from both theorists and experimentalists. It
is generally understood that in order to form Majorana zero modes in
quasi-one-dimensional topological insulators, time-reversal symmetry must be
broken. The straightforward mechanisms for doing so -- applying magnetic fields
or coupling to ferromagnets -- turned out to have many unwanted side effects,
such as degradation of superconductivity and the formation of sub-gap states,
which is part of the reason Majorana zero modes have been eluding direct
experimental detection for a long time. Here we review several proposal that
rely on controlling the phase of the superconducting order parameter, either as
the sole mechanism for time-reversal-symmetry breaking, or as an additional
handy knob used to reduce the applied magnetic field. These proposals hold
practical promise to improve Majorana formation, and they shed light on the
physics underlying the formation of the topological superconducting state.
