A setup with multiple, independent, sequentially monitored data streams is considered. For each of them, two composite hypotheses are postulated, with the interpretation that the stream is anomalous if the corresponding alternative hypothesis holds. It is of interest to detect as quickly as possible whether there is at least one anomalous stream, and also to identify upon stopping the subset of anomalous streams. To address this joint sequential detection and isolation problem, we propose a sequential multiple testing framework where the probabilities of four kinds of error are controlled below distinct, user-specified levels. Two of them refer to the detection task, and the other two to the isolation task. A testing policy is proposed and it is shown to achieve the minimum possible expected sample size, under each point of the parameter space, to a first order asymptotic approximation as the four target error probabilities go to 0. The general theory is illustrated in the case that the data streams generate iid observations that belong to a multiparameter exponential family.