The impressive capabilities of humans to robustly perform manipulation relies on compliant interactions, enabled through the structure and materials spatially distributed in our hands. We propose by mimicking this distributed compliance in an anthropomorphic robotic hand, the open-loop manipulation robustness increases and observe the emergence of human-like behaviours. To achieve this, we introduce the ADAPT Hand equipped with tunable compliance throughout the skin, fingers, and the wrist. Through extensive automated pick-and-place tests, we show the grasping robustness closely mirrors an estimated geometric theoretical limit, while `stress-testing' the robot hand to perform 800+ grasps. Finally, 24 items with largely varying geometries are grasped in a constrained environment with a success rate of 93%. We demonstrate the hand-object self-organization behavior underlines this extreme robustness, where the hand automatically exhibits different grasp types depending on object geometries. Furthermore, the robot grasp type mimics a natural human grasp with a direct similarity of 68%.