We prepare solid-stabilized emulsions using paramagnetic particles at an oil/water interface that can undergo macroscopic phase separation upon application of an external magnetic field. A critical field strength is found for which emulsion droplets begin to translate into the continuous-phase fluid. At higher fields, the emulsions destabilize, leading to a fully phase-separated system. This effect is reversible, and long-term stability can be recovered by remixing the components with mechanical agitation.