We propose a fully-automatic method that computes from scratch point-to-point dense correspondences between isometric shapes under topological noise. While relying on pairwise distance preservation constraints is common and generally sufficient to handle isometric deformations, presence of topological noise needs further actions that we present as our main contributions. First, instead of comparing distances over two paths on two input surfaces, we cast fuzzy votes at the path endpoints based on topologically-robust heat diffusion from path vertices. Second, we make the matching even more stable to topological noise by introducing the so-called reodesics, which are locally shortest geodesics that go through robust matches. In addition to the five standard datasets for isometric shape correspondence with and without topological noise, we employ and release a sixth one geared specifically towards topological noise evaluation with ground-truth information. We demonstrate our qualitative and quantitative advantages over seven recent state-of-the-art methods on these six datasets.