  This work develops the correspondence between orbifolds and free fermion
models. A complete classification is obtained for orbifolds X/G with X the
product of three elliptic curves and G an abelian extension of a group (Z_2)^2
of twists acting on X. Each such quotient X/G is shown to give a geometric
interpretation to an appropriate free fermion model, including the geometric
NAHE+ model. However, the semi-realistic NAHE free fermion model is proved to
be non-geometric: its Hodge numbers are not reproduced by any orbifold X/G. In
particular cases it is shown that X/G can agree with some Borcea-Voisin
threefolds, an orbifold limit of the Schoen threefold, and several further
orbifolds thereof. This yields free fermion models with geometric
interpretations on such special threefolds.
