  We study the adiabatic dynamics of degenerate quantum states induced by loop
paths in a control parameter space. The latter correspond to noisy trajectories
if the system is weakly coupled to environmental modes. On top of conventional
dynamic dephasing, we find a universal non-Abelian geometric dephasing (NAGD)
contribution and express it in terms of the non-Abelian Berry connection and
curvature. We show that NAGD implies either decay or amplification of
coherences as compared to the coherences when only dynamic dephasing is
present. The full NAGD matrix structure can be probed through interference
experiments. We outline such a detection scheme for modified Majorana braiding
setups.
