This article presents a technique for the adaptive refinement of tetrahedral meshes. What makes this method new is that no neighbor information is required for the refined mesh to be compatible everywhere. Refinement consists of inserting new vertices at edge midpoints until some tolerance (geometric or otherwise) is met. For a tetrahedron, the six edges present possible subdivision combinations. The challenge is to triangulate the new vertices (i.e., the original vertices plus some subset of the edge midpoints) in a way that neighboring tetrahedra always generate the same triangles on their shared boundary. A geometric solution based on edge lengths was developed previously, but did not account for geometric degeneracies (edges of equal length). This article provides a solution that works in all cases, while remaining entirely communication-free.