  This talk adapts the available formalism to study a class of heterotic
M-theory vacua with SO(10) grand unification group. Compactification to four
dimensions with N = 1 supersymmetry is achieved on a torus fibered Calabi-Yau
3-fold Z = X / tau_{X} with first homotopy group pi_{1}(Z) = Z_{2}. Here X is
an elliptically fibered Calabi-Yau 3-fold which admits two global sections and
\tau_{X} is a freely acting involution on X. The vacua in this class have net
number of three generations of chiral fermions in the observable sector and may
contain M5-branes in the bulk space which wrap holomorphic curves in Z. Vacua
with nonvanishing and vanishing instanton charges in the observable sector are
considered. The latter case corresponds to potentially viable matter Yukawa
couplings. Since pi_{1}(Z) = Z_{2}, the grand unification group can be broken
with Z_{2} Wilson lines.
  The motivation is to use the above formalism to extend realistic
free-fermionic models to the nonperturbative regime. The correspondence between
these models and Z_{2} x Z_{2} orbifold compactification of the weakly coupled
10-dimensional heterotic string identifies associated Calabi-Yau 3-folds which
possess the structure of the above Z and X. A nonperturbative extension of the
top quark Yukawa coupling is discussed.
