  We present a geometric approach to D-brane model building on the
non-factorisable torus backgrounds of $T^6/\mathbb{Z}_4$, which are $A_3 \times
A_3$ and $A_3 \times A_1 \times B_2$. Based on the counting of `short'
supersymmetric three-cycles per complex structure {\it vev}, the number of
physically inequivalent lattice orientations with respect to the
anti-holomorphic involution ${\cal R}$ of the Type IIA/$\Omega\cal{R}$
orientifold can be reduced to three for the $A_3 \times A_3$ lattice and four
for the $A_3 \times A_1 \times B_2$ lattice. While four independent
three-cycles on $A_3 \times A_3$ cannot accommodate phenomenologically
interesting global models with a chiral spectrum, the eight-dimensional space
of three-cycles on $A_3 \times A_1 \times B_2$ is rich enough to provide for
particle physics models, with several globally consistent two- and
four-generation Pati-Salam models presented here.
  We further show that for fractional {\it sLag} three-cycles, the compact
geometry can be rewritten in a $(T^2)^3$ factorised form, paving the way for a
generalisation of known CFT methods to determine the vector-like spectrum and
to derive the low-energy effective action for open string states.
