  The free fermionic classification method provides a powerful tool to
investigate string vacua, which led to the discovery of spinor--vector duality
and exophobic string models. We extend the classification methodology to both
$\mathcal{N}=1$ and $\mathcal{N}=0$ Flipped $SU(5)$ $\mathbb{Z}_2 \times
\mathbb{Z}_2$ heterotic string orbifolds with asymmetric shifts. The impact of
the asymmetric assignments on the phenomenological characteristics of these
models is investigated. Of particular interest is the analysis of untwisted
moduli fixing for various choices of asymmetric boundary conditions. Two
classes of vacua with different characteristics are systematically investigated
with help from SAT/SMT algorithms, which are shown to increase search
efficiency by up to two orders of magnitude, as well as providing useful tools
to find contradictions between various phenomenological criteria. The general
form of the partition function for the space of models is explained and given
for two specific example models for different choices of asymmetric boundary
conditions. Additionally, the distribution of one-loop cosmological constant
contributions for samples in the two different classes of models are depicted
and discussed.
