  In this work, we study a traversable wormhole sourced by an ideal matter
fluid with an antisymmetric 2-tensor background field coupled to gravity in a
scenario of spontaneously broken Lorentz symmetry. Contrary to employed in the
literature, we use a nonminimal curvature-coupling term
$B^{\mu\nu}B^{\kappa\lambda}R_{\mu\nu\kappa\lambda}$ which incorporates all
three kinds of Lorentz-violating coefficient for the pure-gravity sector of the
minimal standard-model extension. We find that the wormhole is
non-asymptotically globally flat and determine the allowed parameters of the
theory, showing that the matter fluid must be necessarily anisotropic. We also
analyze the energy conditions, checking their validity range and comparing them
with those predicted by general relativity.

General Relativity and Quantum Cosmology