  Birefringent fermions arise as massless fermionic low energy excitations of a
particular tight binding model for spinless fermions on a square lattice which
have two "speeds of light" [M. P. Kennett, ${\it et \, al.}$, Phys. Rev. A
${\bf 83}$, 053636 (2011)]. We use mean field theory to study phases that can
arise when there are nearest neighbour and next-nearest neighbour repulsive
interactions in this model and demonstrate robustness of the birefringent
semi-metal phase in the presence of weak interactions and identify transitions
to staggered density and quantum anomalous Hall ordered phases. We consider the
effect of coupling birefringent fermions to a magnetic field, and find analytic
expressions for the corresponding Landau levels and demonstrate that their
integer Quantum Hall effect displays additional plateaux beyond those observed
for regular Dirac fermions, such as in graphene. We briefly discuss a
tight-binding construction that leads to three dimensional birefringent
fermions.
