Existing learning-based solutions to medical image segmentation have two important shortcomings. First, for most new segmentation task, a new model has to be trained or fine-tuned. This requires extensive resources and machine learning expertise, and is therefore often infeasible for medical researchers and clinicians. Second, most existing segmentation methods produce a single deterministic segmentation mask for a given image. In practice however, there is often considerable uncertainty about what constitutes the correct segmentation, and different expert annotators will often segment the same image differently. We tackle both of these problems with Tyche, a model that uses a context set to generate stochastic predictions for previously unseen tasks without the need to retrain. Tyche differs from other in-context segmentation methods in two important ways. (1) We introduce a novel convolution block architecture that enables interactions among predictions. (2) We introduce in-context test-time augmentation, a new mechanism to provide prediction stochasticity. When combined with appropriate model design and loss functions, Tyche can predict a set of plausible diverse segmentation candidates for new or unseen medical images and segmentation tasks without the need to retrain. Code available at: https://github.com/mariannerakic/tyche/