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For practical bi-directional relaying, symbols transmitted by two sources cannot arrive at the relay with perfect symbol and frame alignments and the asynchronous multiple-access channel (MAC) should be seriously considered. In this paper, we consider the Low-Density Parity-Check (LDPC)-coded BPSK signalling over the general asynchronous MAC with both frame and symbol misalignments. For the symbol-asynchronous MAC, we present a formal log-domain generalized sum-product-algorithm (Log-G-SPA) for efficient decoding. When the frame-asynchronism is encountered at the relay, we propose an original approach by employing the cyclic LDPC codes and the simple cyclic-redundancy-check (CRC) coding technique. Simulation results demonstrate the effectiveness of the proposed approach.