  It has been argued that fluctuations of fermion parity are harmful for the
demonstration of non-Abelian anyonic statistics. Here, we demonstrate a
striking exception in which such fluctuations are actively used. We present a
theory of coherent electron transport from a tunneling tip into a Corbino
geometry Josephson junction where four Majorana bound states (MBSs) rotate.
While the MBSs rotate, electron tunneling happens from the tip to one of the
MBSs thereby changing the fermion parity of the MBSs. The tunneling events in
combination with the rotation allow us to identify a novel braiding operator
that does not commute with the braiding cycles in the absence of tunneling,
revealing the non-Abelian nature of MBSs. The time-averaged tunneling current
exhibits resonances as a function of the tip voltage with a period that is a
direct consequence of the interference between the non-commuting braiding
operations. Our work opens up a possibility for utilizing parity non-conserving
processes to control non-Abelian states.
