In this work, we report on the integrated sensorimotor control of the Pisa/IIT SoftHand, an anthropomorphic soft robot hand designed around the principle of adaptive synergies, with the BRL tactile fingertip (TacTip), a soft biomimetic optical tactile sensor based on the human sense of touch. Our focus is how a sense of touch can be used to control an anthropomorphic hand with one degree of actuation, based on an integration that respects the hand's mechanical functionality. We consider: (i) closed-loop tactile control to establish a light contact on an unknown held object, based on the structural similarity with an undeformed tactile image; and (ii) controlling the estimated pose of an edge feature of a held object, using a convolutional neural network approach developed for controlling other sensors in the TacTip family. Overall, this gives a foundation to endow soft robotic hands with human-like touch, with implications for autonomous grasping, manipulation, human-robot interaction and prosthetics.