Myosin V and VI are antagonistic motors that cohabit membrane vesicles in cells. A systematic study of their collective function, however, is lacking and forms the focus of this study. We functionally reconstitute a two-dimensional actin-myosin interface using myosin V and VI precisely patterned on DNA nanostructures, in combination with a model keratocyte actin meshwork. While scaffolds display solely unidirectional movement, their directional flux is modulated by both actin architecture and the structural properties of the myosin lever arm. This directional flux can be finely-tuned by the relative number of myosin V and VI motors on each scaffold. Pairing computation with experimental observations suggests that the ratio of motor stall forces is a key determinant of the observed competitive outcomes. Overall, our study demonstrates an elegant mechanism for sorting of membrane cargo using equally matched antagonistic motors, simply by modulating the relative number of engagement sites for each motor type.