A new technique has been developed to probe the region of apparent bistability due to a tunneling resonance in the characteristic of a semiconductor asymmetric double-barrier structure. The measuring circuit uses a voltage supply designed to have a load line with positive slope, equivalent to a voltage source and negative series resistance. The appearance of bistability and hysteresis in the characteristic is an artifact of the conventional measuring technique, which employs a load line with negative slope. The complete characteristic is found to be a continuous Z-shaped curve between 50 and 150 K, corresponding to tristability. Equivalent circuit models for the device and voltage supply predict a narrow range of circuit parameters for which a static operating point exists inside the tristable region.