A multiuser detection strategy for coherent demodulation in an asynchronous code-division multiple-access system is proposed and analyzed. The resulting detectors process the sufficient statistics by means of a multistage algorithm based on a scheme for annihilating successive multiple-access interference. An efficient real-time implementation of the multistage algorithm with a fixed decoding delay is obtained and shown to require a computational complexity per symbol which is linear in the number of users K. Hence, the multistage detector contrasts with the optimum demodulator, which is based on a dynamic programming algorithm, has a variable decoding delay, and has a software complexity per symbol that is exponential in K. An exact expression is obtained and used to compute the probability of error is obtained for the two-stage detector, showing that the two-stage receiver is particularly well suited for near-far situations, approaching performance of single-user communications as the interfering signals become stronger. The near-far problem is therefore alleviated. Significant performance gains over the conventional receiver are obtained even for relatively high-bandwidth-efficiency situations.>