Diverse disciplines are interested in how the coordination of interacting agents' movements, emotions, and physiology over time impacts social behavior. Here, we describe a new multivariate procedure for automating the investigation of this kind of behaviorally-relevant 'interactional synchrony', and introduce a novel interactional synchrony measure based on features of dynamic time warping (DTW) paths. We demonstrate that our DTW path-based measure of interactional synchrony between facial action units of two people interacting freely in a natural social interaction can be used to predict how much trust they will display in a subsequent Trust Game. We also show that our approach outperforms univariate head movement models, models that consider participants' facial action units independently, and models that use previously proposed synchrony or similarity measures. The insights of this work can be applied to any research question that aims to quantify the temporal coordination of multiple signals over time, but has immediate applications in psychology, medicine, and robotics.