Schönhardt polyhedron

The Schönhardt polyhedron is an octahedron formed by taking a triangular prism, rotating one triangular face slightly within its plane, and dividing the three now-nonplanar quadrilaterals each into two triangles so that the newly introduced edges are concave. The exact proportions and exact angle of the twist are unimportant, so it is really a family of polyhedra.

It cannot be divided into (finitely many) tetrahedra without introducing new vertices. Rambau showed that this is also true of every "nonconvex twisted prism" where the triangular prism is replaced with an arbitrary convex n-gonal prism.