Global total least squares (GTLS) is a method for the identification of linear systems where no distinction between input and output variables is required. This method has been developed within the deterministic behavioural approach to systems. In this paper we analyse statistical properties of this method when the observations are generated by a multivariable stationary stochastic process. In particular, sufficient conditions for the consistency of GTLS are derived. This means that, when the number of observations tends to infinity, the identified deterministic system converges to the system that provides an optimal appoximation of the data generating process. The two main results are the following. GTLS is consistent if a guaranteed stability bound can be given a priori. If this information is not available, then consistency is obtained if GTLS is applied to the observed data extended with zero values in past and future.