We discuss an experiment in which two magnetic insulators that both show a field-induced magnetic ordering transition are weakly coupled to one another and are placed into an external magnetic field. If the respective magnetic states can be interpreted as phase-coherent Bose–Einstein condensates of magnetic bosonic quasiparticles, one expects the occurrence of Josephson effects. For two identical systems, the resulting d.c. Josephson effect formally represents a constant quasiparticle Josephson current across the weak link, which turns out to be unobservable in an experiment. For magnetic insulators with different critical fields, a spontaneous alternating quasiparticle current develops with a leading oscillation frequency View the MathML source that is determined by the difference between the critical fields. As a result of the coupling, additional sidebands appear in the energy spectrum of the coupled device that would be absent without phase coherence. We discuss the primary conditions for such an effect to take place and conclude that its detection can be feasible for a proper choice of compounds with suitable and realistic material parameters.