The simultaneous robust stabilization of two single-input single-output plants each having additive unstructured uncertainty is considered. First a parametrization of all stabilizing controllers, which simultaneously stabilize the two nominal plants, is developed. A closed-loop stability criterion consisting of an easily checkable necessary condition is then proposed for the defined uncertainty set. Based on this criterion a computationally tractable robust stabilizing compensator design procedure is developed which is applicable to both minimum phase and non-minimum phase nominal plants. Several examples are presented to illustrate the design procedure.