The exact recursion equations in a pair-wise approach are used to study the phase transitions on a two-layer Bethe lattice with crystal field (D) and external magnetic field (H) acting on the layers. The ferromagnetic (FM) and the antiferromagnetic (AFM) interactions for the spins of the upper and the lower layers, respectively, and either a FM- or an AFM-type interaction between the adjacent spins of the layers are assumed. The ground state (GS) phase diagrams of the model are calculated on the (J(2)/J(1), J(3)/qJ(1)) planes for given system parameters, and thirteen distinct GS configurations are obtained. With the GS phase diagrams, the temperature-dependent phase diagrams of the model are obtained by studying the thermal behaviors of the order parameters and the response functions. The model was found to exhibit first- and second-order phase transitions for the coordination numbers q = 3, 4, and 6; hence, tricritical points are also observed. A reentrant behavior is also found whenever the model displays two Neel temperatures.