Compressed sensing (CS) is a signal acquisition paradigm that utilises the finding that a small number of linear projections of a sparse signal have enough information for stable recovery. This paper develops a Bayesian CS algorithm to simultaneously recover multiple signals that follow the Type-3 joint sparse model, where signals share a non-sparse common component and have distinct sparse innovation components. By employing the expectation-maximization (EM) algorithm, the proposed algorithm iteratively updates the estimates of the common component and innovation components. In particular, we find that the update rule for the non-sparse common component in the proposed algorithm, differs from all the other methods in the literature, and we provides an interpretation that gives a valuable insight into why the proposed algorithm is successful in estimating the non-sparse common component. The superior performance of the proposed algorithm is demonstrated by numerical simulation results.