Unlocking the exotic properties promised to occur in topologically non-trivial semi-metals currently requires significant fine-tuning. Crystalline symmetry restricts the location of topological defects to isolated points (0D) or lines (1D), as formalized by the Wigner-Von Neumann theorem. The scarcity of materials in which these anomalies occur at the chemical potential is a major obstacle towards their applications. Here we show how non-crystalline quasi-symmetries stabilize near-degeneracies of bands over extended regions in energy and in the Brillouin zone. Specifically, a quasi-symmetry is an exact symmetry of a k∙p Hamiltonian to lower-order that is broken by higher-order terms. Hence quasi-symmetric points are gapped, yet the gap is parametrically small and therefore does not influence the physical properties of the system. We demonstrate that in the eV-bandwidth semi-metal CoSi an internal quasi-symmetry stabilizes gaps in the 1-2 meV range over a large near-degenerate plane (2D). This quasi-symmetry is key to explaining the surprising simplicity of the experimentally observed quantum oscillations of four interpenetrating Fermi surfaces around the R-point. Untethered from the limitations of crystalline symmetry, quasi-symmetries eliminate the need for fine-tuning as they enforce sources of large Berry curvature to occur at the chemical potential, and thereby lead to new Wigner-Von Neumann classifications of solids. Quasi-symmetries arise from a comparable splitting of degenerate states by spin-orbit coupling and by orbital dispersion - suggesting a hidden classification framework for symmetry groups and materials in which quasi-symmetries are critical to understand the low-energy physics.