The ground breaking extension of the key concept of phase structure to nonequilibrium regimes was only recently achieved in Floquet systems, characterized by a time-dependent quantum Hamiltonian with a periodic driving source. However, despite the theoretical advances, only very few systems are known to display experimental Floquet phases, not one of them employing a laser emission-based mechanism. Here we report the first experimental observation of a Floquet phase in a photonic system, a disordered fiber laser with spatial eigenmode localization. We apply a periodically oscillating cw pumping source that drives the random couplings of the Floquet Hamiltonian. A photonic Floquet spin-glass phase is demonstrated in the random-lasing regime by extensive measurements of the Parisi overlap parameter and asymmetry properties of its distribution. In contrast, in the fluorescent regime below threshold, the absence of mode localization prevents the stabilization of a Floquet phase. Our results are nicely described by theoretical arguments.