  We study the phase diagram of an effective model of competing nematic and
antiferromagnetic orders of interacting electrons on the Bernal-stacked
honeycomb bilayer, as relevant for bilayer graphene. In the noninteracting
limit, the model features a semimetallic ground state with quadratic band
touching points at the Fermi level. Taking the effects of short-range
interactions into account, we demonstrate the existence of an extended region
in the mean-field phase diagram characterized by coexisting nematic and
antiferromagnetic orders. By means of a renormalization group approach, we
reveal that the quantum phase transition from nematic to coexistent
nematic-antiferromagnetic orders is continuous and characterized by emergent
Lorentz symmetry. It falls into the $(2+1)$-dimensional relativistic
Gross-Neveu-Heisenberg quantum universality class, which has recently been much
investigated in the context of interacting Dirac systems in two spatial
dimensions. The coexistence-to-antiferromagnetic transition, by contrast, turns
out to be weakly first order as a consequence of the absence of continuous
spatial rotational symmetry on the honeycomb bilayer. Implications for
experiments in bilayer graphene are discussed.
