This paper addresses the following general scenario: a scientist wishes to perform a battery of experiments, each generating a sequential stream of data, to investigate some phenomenon. The scientist would like to control the overall error rate in order to draw statistically valid conclusions from each experiment, while being as efficient as possible. The between-stream data may differ in distribution and dimension but also may be highly correlated, even duplicated exactly in some cases. Treating each experiment as a hypothesis test and adopting the familywise error rate (FWER) metric, we give a procedure that sequentially tests each hypothesis while controlling both the type I and II FWERs regardless of the between-stream correlation, and only requires arbitrary sequential test statistics that control the error rates for a given stream in isolation. The proposed procedure, which we call the sequential Holm procedure because of its inspiration from Holm׳s (1979) seminal fixed-sample procedure, shows simultaneous savings in expected sample size and less conservative error control relative to fixed sample, sequential Bonferroni, and other recently proposed sequential procedures in a simulation study.