  The one-step replica symmetry breaking (RSB) is used to study a
two-sublattice fermionic infinite-range Ising spin glass (SG) model in a
transverse field $\Gamma$. The problem is formulated in a Grassmann path
integral formalism within the static approximation. In this model, a parallel
magnetic field $H$ breaks the symmetry of the sublattices. It destroys the
antiferromagnetic (AF) order, but it can favor the nonergodic mixed phase
(SG+AF) characterizing an asymmetric RSB region. In this region,
intra-sublattice disordered interactions $V$ increase the difference between
the RSB solutions of each sublattice. The freezing temperature shows a higher
increase with $H$ when $V$ enhances. A discontinue phase transition from the
replica symmetry (RS) solution to the RSB solution can appear with the presence
of an intra-sublattice ferromagnetic average coupling. The $\Gamma$ field
introduces a quantum spin flip mechanism that suppresses the magnetic orders
leading them to quantum critical points. Results suggest that the quantum
effects are not able to restore the RS solution. However, in the asymmetric RSB
region, $\Gamma$ can produce a stable RS solution at any finite temperature for
a particular sublattice while the other sublattice still presents RSB solution
for the special case in which only the intra-sublattice spins couple with
disordered interactions.
