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In this paper, we develop a statistical approach based on Markov chain Monte Carlo (MCMC) techniques for joint data detection and channel estimation over time-varying frequency-selective channels. The proposed detector, that we call MCMC with list channel estimates (MCMC-LCE), adopts the Gibbs sampler to find a list of mostly likely transmitted sequences and matching channel estimates/impulse responses (CIR), to compute the log-likelihood ratio (LLR) of transmitted bits. The MCMC-LCE provides a low-complexity means to approximate the optimal maximum a posterior (MAP) detection in a statistical fashion and is applicable to channels with long memory. Promising behavior of the MCMC-LCE is presented using both synthetic channels and real data collected from underwater acoustic (UWA) channels whose large delay spread and time variation have been the main motivation for the developed system. We also adopt an adaptive variable step-size least mean-square (VSLMS) algorithm for channel tracking. We find that this choice, which does not require prior knowledge on the CIR statistics, is a good fit for UWA channels. Superior performance of the MCMC-LCE over turbo minimum mean-square-error (MMSE) equalizers is demonstrated for a variety of channels examined in this work.