Beam self-cleaning in multimode optical fibers has recently been interpreted within a thermodynamic framework, in which nonlinear propagation drives the optical field towards a statistical equilibrium state. While this interpretation has been supported by measurements of modal populations and entropy, the role of shot-to-shot spatial fluctuations of the output beam has so far remained unexplored. Here, we experimentally show that the transverse output beam profile becomes progressively more reproducible from shot to shot as beam self-cleaning develops. This behavior is consistent with the thermodynamic expectation that relative fluctuations become increasingly small as the system approaches equilibrium.