  In this paper we consider two different nonlinear $\sigma$-models minimally
coupled to Eddington-inspired Born-Infeld gravity. We show that the resultant
geometries represent minimal modifications with respect to those found in GR,
though with important physical consequences. In particular, wormhole structures
always arise, though this does not guarantee by itself the geodesic
completeness of those space-times. In one of the models, quadratic in the
canonical kinetic term, we identify a subset of solutions which are regular
everywhere and are geodesically complete. We discuss characteristic features of
these solutions and their dependence on the relationship between mass and
global charge.
