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This paper describes a new approach to list-based soft-input soft-output (SISO) decoding based on order-i reprocessing. Approximations to both the log-maximum a posteriori (MAP) and max-log-MAP algorithms are developed. Additional decoding steps are proposed to correct common types of errors remaining after iterative decoding. These steps can significantly improve performance at low bit-error rates in later iterations. The proposed algorithms offer a wide range of complexity versus performance tradeoffs, which are explored through Monte Carlo simulations of product code decodings. The algorithms improve performance over previous approaches.