This paper presents a control scheme for force sensitive, gentle grasping with a Pisa/IIT anthropomorphic SoftHand equipped with a miniaturised version of the TacTip optical tactile sensor on all five fingertips. The tactile sensors provide high-resolution information about a grasp and how the fingers interact with held objects. We first describe a series of hardware developments for performing asynchronous sensor data acquisition and processing, resulting in a fast control loop sufficient for real-time grasp control. We then develop a novel grasp controller that uses tactile feedback from all five fingertip sensors simultaneously to gently and stably grasp 43 objects of varying geometry and stiffness, which is then applied to a human-to-robot handover task. These developments open the door to more advanced manipulation with underactuated hands via fast reflexive control using high-resolution tactile sensing.