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Within Dialogue Modeling research in AI and NLP, considerable attention has been spent on “dialogue state tracking” (DST), which is the ability to update the representations of the speaker’s needs at each turn in the dialogue by taking into account the past dialogue moves and history. Less studied but just as important to dialogue modeling, however, is “common ground tracking” (CGT), which identifies the shared belief space held by all of the participants in a task-oriented dialogue: the task-relevant propositions all participants accept as true. In this paper we present a method for automatically identifying the current set of shared beliefs and “questions under discussion” (QUDs) of a group with a shared goal. We annotate a dataset of multimodal interactions in a shared physical space with speech transcriptions, prosodic features, gestures, actions, and facets of collaboration, and operationalize these features for use in a deep neural model to predict moves toward construction of common ground. Model outputs cascade into a set of formal closure rules derived from situated evidence and belief axioms and update operations. We empirically assess the contribution of each feature type toward successful construction of common ground relative to ground truth, establishing a benchmark in this novel, challenging task.