A magnon spin-orbit coupling, induced by the dipole-dipole interaction, is derived in monoclinic-stacked bilayer honeycomb spin lattice with perpendicular magnetic anisotropy and antiferromagnetic interlayer coupling. Linear crossings are predicted in the magnon spectrum around the band minimum in valley, as well as in the high-frequency range around the zone boundary. The linear crossings in K and K valleys, which connect the acoustic and optical bands, can be gapped when the intralayer dipole-dipole or Kitaev interactions exceed the interlayer dipole-dipole interaction, resulting in a phase transition from semimetal to insulator. Our results are useful for analyzing the magnon spin dynamics and transport properties in van der Waals antiferromagnets.