The identification of fractionalized excitations, such as Majorana quasi-particles, would be a striking signal of the realization of exotic quantum states of matter. While the paramount demonstration of such excitations would be a probe of their non-Abelian statistics via controlled braiding operations, alternative proposals exist that may be easier to access experimentally. Here we identify a signature of Majorana quasi-particles, qualitatively different from the behaviour of a conventional superconductor, which can be detected in cold atom systems using alkaline-earth-like atoms. The system studied is 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-equilibrium Josephson effect with a characteristic 8π periodicity of the Josephson current. The visibility of the 8π periodicity of the Josephson current is then studied including the effects of dephasing and particle losses.