We describe the phase behavior of an aqueous mixture of discotic nanoparticles of laponite and spherical magnetic nanoparticles of maghemite. To obtain stable mixtures from a chemical point of view, the maghemite nanoparticles are first coated by a thin layer of silica in order to adapt their surface chemistry to that of laponite nanoparticles: this enables one to raise volume fractions of maghemite Phi mag in the laponite suspensions up to several percent. Although the system is out of equilibrium, a "fluid-solid" state diagram was established showing that the mixtures undergo a fluid-solid transition, similar to that of pure suspensions of laponite, over a given volume fraction of laponite Phi lap and over a given Phi mag. An increase in Phi mag shifts Phi lap toward the lower values. When a solid sample is just above Phi lap, the application of an external magnetic field gradient induces a solid-to-liquid transition if the sample is located not too far from Phi lap on the state diagram. The structure of the mixtures, determined either at small scale by small-angle neutron scattering (SANS) or at intermediate scales by optical microscopy, shows that the solid samples are phase separated at a local scale: they are made of densely connected domains of laponite nanoparticles surrounding liquid pockets of maghemite nanoparticles. The size of the pockets grows with time. The magnetic liquid pockets are responsible for the rupture of the solid samples when an external magnetic field gradient is applied since their deformation induces local mechanical stress, internally damaging the network formed by the solid domains of laponite. The microscopic phase separation is the result of two opposite effects: (i) entropic effects that tend to phase separate the system macroscopically when the packing entropy overcomes the orientational entropy and (ii) long-range electrostatic repulsions that freeze the system.