Non-rigid 3D mesh matching is a critical step in computer vision and computer graphics pipelines. We tackle matching meshes that contain topological artefacts which can break the assumption made by current approaches. While Functional Maps assume the deformation induced by the ground truth correspondences to be near-isometric, ARAP-like deformation-guided approaches assume the latter to be ARAP. Neither assumption holds in certain topological configurations of the input shapes. We are motivated by real-world scenarios such as per-frame multi-view reconstructions, often suffering from topological artefacts. To this end, we propose a topology-adaptive deformation model allowing changes in shape topology to align shape pairs under ARAP and bijective association constraints. Using this model, we jointly optimise for a template mesh with adequate topology and for its alignment with the shapes to be matched to extract correspondences. We show that, while not relying on any data-driven prior, our approach applies to highly non-isometric shapes and shapes with topological artefacts, including noisy per-frame multi-view reconstructions, even outperforming methods trained on large datasets in 3D alignment quality.