Tactile sensing is required for the dexterous manipulation of objects in robotic applications. In particular, the ability to measure and distinguish in real time normal and shear forces is crucial for slip detection and interaction with fragile objects. Here, we report a biomimetic soft electronic skin (e-skin) that is composed of an array of capacitors and capable of measuring and discriminating in real time both normal and tangential forces. It is enabled by a three-dimensional structure that mimics the interlocked dermis-epidermis interface in human skin. Moreover, pyramid microstructures arranged along nature-inspired phyllotaxis spirals resulted in an e-skin with increased sensitivity, minimal hysteresis, excellent cycling stability, and response time in the millisecond range. The e-skin provided sensing feedback for controlling a robot arm in various tasks, illustrating its potential application in robotics with tactile feedback.