The problem of detecting and isolating a correlated pair among multiple Gaussian information sources is considered. It is assumed that there is at most one pair of correlated sources and that observations from all sources are acquired sequentially. The goal is to stop sampling as quickly as possible, declare upon stopping whether there is a correlated pair or not, and if yes, to identify it. Specifically, it is required to control explicitly the probabilities of three kinds of error: false alarm, missed detection, wrong identification. We propose a procedure that not only controls these error metrics, but also achieves the smallest possible average sample size, to a first-order approximation, as the target error rates go to 0. Finally, a simulation study is presented in which the proposed rule is compared with an alternative sequential testing procedure that controls the same error metrics.