Since its first demonstration, spatial beam self-cleaning has been targeted as a breakthrough nonlinear effect, for its potential of extending to multimode fibers different technologies based on single-mode fibers, such as fiber lasers and endoscopes. To date, most of the theoretical descriptions of beam self-cleaning are based on scalar models. Whereas, in experiments the analysis of the polarization state of self-cleaned beams is often neglected. Here, we fill this gap between theory and experiments, by demonstrating that a self-cleaned beam eventually loses its degree of polarization, as long as linearly polarized light of enough power is injected at the fiber input. Our results are cast in the framework of a thermodynamic description of nonlinear beam propagation in multimode fibers, providing the first experimental proof of the applicability of scalar theories for the description of the spatial beam self-cleaning effect.