This paper develops closed-loop tactile controllers for dexterous robotic manipulation with a dual-palm robotic system. Tactile dexterity is an approach to dexterous manipulation that plans for robot/object interactions that render interpretable tactile information for control. We divide the role of tactile control into two goals: 1) control the contact state between the end-effector and the object (contact/no-contact, stick/slip) by regulating the stability of planned contact configurations and monitoring undesired slip events; and 2) control the object state by tactile-based tracking and iterative replanning of the object and robot trajectories. Key to this formulation is the decomposition of manipulation plans into sequences of manipulation primitives with simple mechanics and efficient planners. We consider the scenario of manipulating an object from an initial pose to a target pose on a flat surface while correcting for external perturbations and uncertainty in the initial pose of the object. We experimentally validate the approach with an ABB YuMi dual-arm robot and demonstrate the ability of the tactile controller to react to external perturbations.