  We construct a specific example of a class of traversable wormholes in
Einstein-Dirac-Maxwell theory in four spacetime dimensions, without needing any
form of exotic matter. Restricting to a model with two massive fermions in a
singlet spinor state, we show the existence of spherically symmetric
asymptotically flat configurations which are free of singularities,
representing localized states. These solutions satisfy a generalized Smarr
relation, being connected with the extremal Reissner-Nordstr\"om black holes.
They also possess a finite mass $M$ and electric charge $Q_e$, with $Q_e/M>1$.
An exact wormhole solution with ungauged, massless fermions is also reported.
