Grip control during robotic in-hand manipulation is usually modeled as part of a monolithic task, relying on complex controllers specialized for specific situations. Such approaches do not generalize well and are difficult to apply to novel manipulation tasks. Here, we propose a modular object stabilization method based on a proposition that explains how humans achieve grasp stability. In this bio-mimetic approach, independent tactile grip stabilization controllers ensure that slip does not occur locally at the engaged robot fingers. Such local slip is predicted from the tactile signals of each fingertip sensor i.e., BioTac and BioTac SP by Syntouch. We show that stable grasps emerge without any form of central communication when such independent controllers are engaged in the control of multi-digit robotic hands. These grasps are resistant to external perturbations while being capable of stabilizing a large variety of objects.