A challenging and important problem for tendon-driven multi-fingered robotic hands is to ensure grasping adaptivity while minimizing the number of actuators needed to provide human-like functionality. Inspired by the Pisa/IIT SoftHand, this paper introduces a 3D-printed, highly-underactuated, tactile-sensorized, five-finger robotic hand named the Tactile SoftHand-A, which features an antagonistic mechanism to actively open and close the hand. Our proposed dual-tendon design gives options that allow active control of specific (distal or proximal interphalangeal) joints; for example, to adjust from an enclosing to fingertip grasp or to manipulate an object with a fingertip. We also develop and integrate a new design of fully 3D-printed vision-based tactile sensor within the fingers that requires minimal hand assembly. A control scheme based on analytically extracting contact location and slip from the tactile images is used to coordinate the antagonistic tendon mechanism (using a marker displacement density map, suitable for TacTip-based sensors). We perform extensive testing of a single finger, the entire hand, and the tactile capabilities to show the improvements in reactivity, load-bearing, and manipulability in comparison to a SoftHand that lacks the antagonistic mechanism. We also demonstrate the hand's reactivity to contact disturbances including slip, and how this enables teleoperated control from human hand gestures. Overall, this study points the way towards a class of low-cost, accessible, 3D-printable, tactile, underactuated human-like robotic hands, and we openly release the designs to facilitate others to build upon this work. The designs are open-sourced at https://github.com/HaoranLi-Data/Tactile_SoftHand_A