The objective of this paper is to present a Bayesian classification technique that can be used to classify a multivariate time series realization into one of several multivariate autoregressive sources. The main assumption here is that the sources share a common unknown order. This is to be closer to the real situations, whereas the order of a process is usually unknown or at least, has to be estimated. Hence, the order of the processes will be regarded as a nuisance parameter that has a maximum known value. The proposed technique is based on deriving the marginal posterior mass function of a classification vector, then one can assign the multivariate realization to the r-th multivariate source whenever the posterior mass function of the classification vector has its largest value at the r-th mass point. A simulation technique is carried out to study the numerical efficiency of the proposed classification technique. The simulation studies the behavior of the technique with respect to time series length, the maximum value assumed to the order of the process, and the parameter values, on the performance of the proposed technique.