If a mobile computing device knows how it is positioned and oriented in relation to other devices nearby, then it can provide enhanced support for multi-device and multi-user interactions. Existing systems that provide position information to mobile computers are reliant on externally deployed infrastructure, such as beacons or sensors in the environment. We introduce the Relate system, which provides fine-grained relative position information to co-located devices on the basis of peer-to-peer sensing, thus overcoming dependence on any external infrastructure. The system is realised as a hardware/software plug-in, using ultrasound for peer-to-peer sensing, USB to interface with standard mobile devices, and data abstraction and inferencing to map sensor data to a spatial model that maintains both quantitative and qualitative relationships. We present a set of services and applications to demonstrate the utility of the system. We report experimental results on the accuracy of the relative position and orientation estimates, and other aspects of system performance.