The choice of the Fermi energy in the electrodes of a GaSb/AlSb/InAs/AlSb/GaSb type-II heterostructure has deep consequences on the physics of interband tunneling. We show that tunneling between electron and hole bands will result in three different current-voltage characteristics depending on the Fermi energy in the electrodes of the device. When electrode doping is high, the device characteristics exhibit a current shoulder at low bias followed by a current peak. Magnetotunneling experiments reveal additional current satellites: one emerging from the shoulder and another one emerging from the peak. The shift in the voltage position of these two features and that of the peak current is opposite when the magnetic field increases. This splitting is evidence of a hybridization gap due to k⋅p coupling between the electron band in InAs and the hole bands in GaSb.