  We calculate the energy spectrum of a Dirac double layer, where each layer
has the Dirac electronic dispersion, in the presence of a tilted magnetic field
and small interlayer tunneling. We show that the energy splitting between the
Landau levels has an oscillatory dependence on the in-plane magnetic field and
vanishes at a series of special tilt angles of the magnetic field. Using a
semiclassical analysis, we show that these special tilt angles are determined
by the Berry phase of the Dirac Hamiltonian. The interlayer tunneling
conductance also exhibits an oscillatory dependence on the magnetic field tilt
angle, known as the angular magnetoresistance oscillations (AMRO). Our results
are applicable to graphene double layers and thin films of topological
insulators.
