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The ability to adapt medical image segmentation networks for a novel class such as an unseen anatomical or pathological structure, when only a few labelled examples of this class are available from local healthcare providers, is sought-after. This potentially addresses two widely recognised limitations in deploying modern deep learning models to clinical practice, expertise-and-labour-intensive labelling and cross-institution generalisation. This work presents the first 3D few-shot interclass segmentation network for medical images, using a labelled multi-institution dataset from prostate cancer patients with eight regions of interest. We propose an image alignment module registering the predicted segmentation of both query and support data, in a standard prototypical learning algorithm, to a reference atlas space. The built-in registration mechanism can effectively utilise the prior knowledge of consistent anatomy between subjects, regardless whether they are from the same institution or not. Experimental results demonstrated that the proposed registration-assisted prototypical learning significantly improved segmentation accuracy (p-values<0.01) on query data from a holdout institution, with varying availability of support data from multiple institutions. We also report the additional benefits of the proposed 3D networks with 75% fewer parameters and an arguably simpler implementation, compared with existing 2D few-shot approaches that segment 2D slices of volumetric medical images.