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Autonomous surgical robots are a promising solution to the increasing demand for surgery amid a shortage of surgeons. Recent work has proposed learning-based approaches for the autonomous manipulation of soft tissue. However, due to variability in tissue geometries and stiffnesses, these methods do not always perform optimally, especially in out-of-distribution settings. We propose, develop, and test the first application of uncertainty quantification to learned surgical soft-tissue manipulation policies as an early identification system for task failures. We analyze two different methods of uncertainty quantification, deep ensembles and Monte Carlo dropout, and find that deep ensembles provide a stronger signal of future task success or failure. We validate our approach using the physical daVinci Research Kit (dVRK) surgical robot to perform physical soft-tissue manipulation. We show that we are able to successfully detect out-of-distribution states leading to task failure and request human intervention when necessary while still enabling autonomous manipulation when possible. Our learned tissue manipulation policy with uncertainty-based early failure detection achieves a zero-shot sim2real performance improvement of 47.5% over the prior state of the art in learned soft-tissue manipulation. We also show that our method generalizes well to new types of tissue as well as to a bimanual soft-tissue manipulation task.