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.
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