To fully exploit sea experiments under controlled and reproducible laboratory conditions, a channel model driven by real data is derived. This model relies on the assumption that a channel recorded at sea is a single observation of an underlying random process. From this single observation, the channel statistical properties are estimated to then feed a stochastic simulator that generates multiple realizations of the underlying process. Based on the analysis of data collected in the Atlantic Ocean and the Mediterranean Sea, we fully relax the usual wide-sense stationary uncorrelated scattering (WSSUS) assumption. We show thanks to the empirical mode decomposition that a trend stationary model suits the analyzed underwater acoustic communication channels very well. Scatterers with different path delays are also assumed to be potentially correlated so that the true second order statistics of the channel are taken into account by our model. Test cases illustrate the benefits of channel stochastic replay to communication system design and validation. The Matlab code corresponding to the proposed simulator is available at http://perso.telecom-bretagne.eu/fxsocheleau/software.