  We theoretically study a Kitaev wire interrupted by an extra site which gives
rise to super exchange coupling between two Majorana bound states. We show that
this system hosts a tunable, non-equlibrium Josephson effect with a
characteristic $8\pi$ periodicity of the Josephson current. We elucidate the
physical mechanism deriving a minimal model for the junction and confirm its
quantitative accuracy by comparison to the numerical solution of the full
model. The visibility of the $8\pi$ periodicity of the Josephson current is
then studied using time-dependent simulations including the effects of
dephasing and particle losses. Our findings provide a novel signature of
Majorana quasi-particles which is qualitatively different form the behavior of
a conventional superconductor, and can be experimentally verified in cold atom
systems using alkaline-earth-like atoms.
