  Using the transfer matrix method, we numerically investigate the structure of
spatial coherences and their fluctuations in the 3d Anderson model in the
metallic phase when driven out-of-equilibrium by external leads at zero
temperature and in linear response. We find that the stationary state entails
non-local non-Gaussian correlations in the longitudinal direction, which are
characteristic of diffusive non equilibrium steady states. These correlations
are quantitatively matched, at least up to third order, by those analytically
derived in the Quantum Symmetric Simple Exclusion Process (QSSEP) which
describes diffusive fermions in 1d subject to dynamical disorder. Furthermore,
the large deviation scaling and $U(1)$ invariance of these correlations imply a
link between the Anderson model and free probability theory. Our findings
suggest the existence of a universal structure of correlations in
non-interacting diffusive quantum systems that might be captured by QSSEP.
