  The liquid-to-ordered phase transition in a bilayer system of fermions is
studied within the context of a recently proposed density-functional theory
[Phys. Rev. A {\bf 92}, 023614 (2015)]. In each two-dimensional layer, the
fermions interact via a repulsive, isotropic dipolar interaction. The presence
of a second layer introduces an attractive {\em interlayer} interaction,
thereby allowing for inhomogeneous density phases which would otherwise be
energetically unfavourable. For any fixed layer separation, we find an
instability to a commensurate one-dimensional stripe phase in each layer, which
always precedes the formation of a triangular Wigner crystal. However, at a
certain {\em fixed} coupling, tuning the separation can lead to the system
favoring a commensurate triangular Wigner crystal, or one-dimensional stripe
phase, completely bypassing the Fermi liquid state. While other crystalline
symmetries, with energies lower than the liquid phase can be found, they are
never allowed to form owing to their high energetic cost relative to the
triangular Wigner crystal and stripe phase.
