  We consider D=6, N=1, Z_M orbifold compactifications of heterotic strings in
which the usual modular invariance constraints are violated. It is argued that
in the presence of non-perturbative effects many of these vacua are
nevertheless consistent. The perturbative massless sector can be computed
explicitly from the perturbative mass formula subject to an extra shift in the
vacuum energy. This shift is associated to a non-trivial antisymmetric B-field
flux at the orbifold fixed points. The non-perturbative piece is given by
five-branes either moving in the bulk or stuck at the fixed points, giving rise
to Coulomb phases with tensor multiplets. The heterotic duals of some Type IIB
orientifolds belong to this class of orbifold models. We also discuss how to
carry out this type of construction to the D=4, N=1 case and specific
$Z_M\times Z_M$ examples are presented in which non-perturbative transitions
changing the number of chiral generations do occur.

High Energy Physics - Theory;High Energy Physics - Phenomenology