Physical layer cooperation of a source with a relay can significantly boost the performance of a wireless connection. However, the best practical relaying scheme can vary depending on the relative strengths of the channels that connect the source, relay and destination. This paper proposes and evaluates QUILT, a system for physical-layer relaying that seamlessly adapts to the underlying network configuration to achieve competitive or better performance as compared to the best current approaches. QUILT combines on-demand, opportunistic use of Decode-Forward (DF) or Quantize-Map-Forward (QMF) followed by interleaving at the relay, with hybrid decoding at the destination that extracts information of received frames even if these are not decodable. We theoretically quantify how our design choices for QUILT affect the system's performance. We also deploy QUILT on the WarpLab software radio platform, and show through over the over-the-air experiments up to 5 times FER improvement over the next best cooperative protocol.