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We present a joint channel estimation and decoding method for space-time trellis codes (STTC) transmitted over an unknown flat fading multiple-input multiple-output (MIMO) channel. In a blind context, i.e. when pilot symbols are not available, we will show that the probability density function (pdf) of the MIMO channel becomes multimodal due to phase ambiguities. To address this problem, a combined state-space model for the time-varying MIMO channel and the STTC is introduced. The proposed algorithm then uses a Gaussian mixture approach to perform joint channel estimation and soft-output detection. We compare our method to several well known algorithms obtained by collapsing the Gaussian mixture to a single Gaussian distribution. A comparison with sequential Monte Carlo methods is also included. Numerical simulations show that the performances of the proposed blind STTC decoder are close to decoding with perfect channel state information (CSI). As a useful application of the proposed soft-output algorithm, we provide an example of blind iterative decoding of concatenated space-time codes.