  Bose-Einstein condensation (BEC) of triplet excitations triggered by a
magnetic field, sometimes called magnon BEC, in dimerized antiferromagnets
gives rise to a long-range antiferromagnetic order in the plane perpendicular
to the applied magnetic field. To explore the effects of spin-orbit coupling on
magnon condensation, we study a spin model on a distorted honeycomb lattice
with dimerized Heisenberg exchange ($J$ terms) and uniform off-diagonal
exchange ($\Gamma'$ terms) interactions. Via variational Monte Carlo
calculations and spin-wave theory, we find that an out-of-plane magnetic field
can induce different types of long-range magnetic orders, no matter if the
ground state is a non-magnetic dimerized state or an ordered N\'{e}el state.
Furthermore, the critical properties of field-driven phase transitions in the
presence of spin-orbit couplings, as illustrated from spin-wave spectrum and
interpreted by effective field theory, can be different from the conventional
magnon BEC. Our study is helpful to understand the rich phases of spin-orbit
coupled antiferromagnets induced by magnetic fields.
