Many arthropods and small vertebrates use adhesive pads for climbing. These biological adhesives have to meet conflicting demands: attachment must be strong and reliable, yet detachment should be fast and effortless. Climbing animals can rapidly and reversibly control their pads' adhesive strength by shear forces, but the mechanisms underlying this coupling have remained unclear. Here, we show that adhesive forces of stick insect pads closely followed the predictions from tape peeling models when shear forces were small, but strongly exceeded them when shear forces were large, resulting in an approximately linear increase of adhesion with friction. Adhesion sharply increased at peel angles less than ca 30°, allowing a rapid switch between attachment and detachment. The departure from classic peeling theory coincided with the appearance of pad sliding, which dramatically increased the peel force via a combination of two mechanisms. First, partial sliding pre-stretched the pads, so that they were effectively stiffer upon detachment and peeled increasingly like inextensible tape. Second, pad sliding reduces the thickness of the fluid layer in the contact zone, thereby increasing the stress levels required for peeling. In combination, these effects can explain the coupling between adhesion and friction that is fundamental to adhesion control across all climbing animals. Our results highlight that control of adhesion is not solely achieved by direction-dependence and morphological anisotropy, suggesting promising new routes for the development of controllable bio-inspired adhesives.