A key issue in the two-path successive relaying (TPSR) communications systems is the interference, which is caused by the concurrent transmission of the source and one of two relays. The previously reported investigations show that interference cancellation is required at the relays and destination to avoid severe performance degradation. In this work, we introduce a different approach that does not require interference compensation for decode-and-forward TPSR systems, while maintaining diversity gain and full rate. At the listening relay, the proposed approach enables the separation between the source and the forwarding relay transmissions by using two carriers of the same frequency but in phase quadrature. Therefore, a maximum-likelihood data detector can be adopted without being interfered by the inter-relay link. At the destination node, we make use of the interference signal to develop optimal and sub-optimal data detectors. Furthermore, we derive a theoretical lower bound on the bit-error-rate performance. Simulation and analytical results demonstrate the superiority of the proposed approach compared to the conventional schemes reported in the literature.