As global demand for fruits and vegetables continues to rise, the agricultural industry faces significant challenges in securing adequate labor. Robotic harvesting devices offer a promising solution to address this issue. Harvesting delicate fruits, such as blackberries, presents unique open challenges due to their fragility. This paper introduces BerryTwist, a prototype robotic gripper specifically designed for blackberry harvesting. The gripper features a fabric tube mechanism that uses motorized twisting action to gently envelop the fruit, ensuring uniform pressure application and minimizing damage. The twisting motion is transferred to the tube through a compliant mechanism, thus maintaining the overall softness of the structure. We thoroughly tested BerryTwist, paying particular attention to the effect of varying tube properties. We developed three types of tubes varying in elasticity and compressibility, using foam padding, spandex, and food-safe cotton cheesecloth. Performance testing focused on assessing each gripper's ability to detach and release blackberries, with an emphasis on quantifying damage rates. The results indicate that the proposed gripper achieved an 82% success rate in detaching blackberries and a 95% success rate in releasing them, demonstrating its promising potential for robotic harvesting applications. Finally, we will demonstrate the robotic harvesting operation by establishing a simple farm setup and integrating the gripper with Franka Emika's robot manipulator.