  The Einstein-Maxwell-Klein-Gordon Lagrangian is supplemented by a non-minimal
coupling of the real scalar field to the Gauss-Bonnet invariant. The non
minimal coupling function is chosen as a general second degree polynomial in
the scalar field for which the system is known to admit hairy black holes. The
new interaction leads naturally to a violation of the null energy condition,
allowing for wormholes to exist without the need of exotic matter.
  Spherically symmetric, charged wormholes are constructed and their domain of
existence is determined in terms of the different choices of the non-minimal
coupling constants and of the electric charge. A special emphasis is set to the
case of the purely quadratic coupling function.
  A phenomenon reminiscent to spontaneously scalarised black holes occurs for
wormholes. The interaction with the electromagnetic field leads to new families
of wormholes supported by a non-vanishing, large enough, electric charge.
