  The interaction between microwave photons and magnons is well understood and
originates from the Zeeman coupling between spins and a magnetic field.
Interestingly, the magnon/photon interaction is accompanied by a phase factor
which can usually be neglected. However, under the rotating wave approximation,
if two magnon modes simultaneously couple with two cavity resonances, this
phase cannot be ignored as it changes the physics of the system. We consider
two such systems, each differing by the sign of one of the magnon/photon
coupling strengths. This simple difference, originating from the various
coupling phases in the system, is shown to preserve, or destroy, two potential
applications of hybrid photon/magnon systems, namely dark mode memories and
cavity-mediated coupling. The observable consequences of the coupling phase in
this system is akin to the manifestation of a discrete Pancharatnam-Berry
phase, which may be useful for quantum information processing.
