We consider electrochemical cells with co-ionic solid electrolyte membranes and analyze charge transport in the membrane and multiple electrochemical reactions at the triple-phase boundaries. Charged-species concentrations at the membrane boundaries are dictated by fast non-electrochemical reactions with the gas-phase oxygen, hydrogen, and water vapor, which endow the membrane with its co-ionic conductivity properties. The electrochemical reactions are described in terms of the Butler-Volmer equation. Charge conservation is invoked to couple the rates of transport within the membrane and the rates of the electrochemical reactions and close the model system of equations. The electrical potentials of electrodes and membranes at their boundary are unique, and hence the over potentials of different reactions cannot in general be simultaneously zero. For a single pair of half-cell reactions, the model behavior is as expected, regarding the direction of reactions, the effect of applied voltage, and the effect of kinetic constants on overpotentials. When multiple reactions take place, depending on their relative speeds, they may collaborate or compete to determine the net charge flux. The fastest reactions determine the local membrane electrical potential and, hence, the sign and magnitude of the overpotentials of other reactions, and thus their direction.