Apparently such a space exists, and apparently Milnors solution to Hauptvermutung also explains this situation. I am learning about topologically stratified spaces, so it would be nice to know a reference of a situation when the link is not necessarily well defined.
There is a reference to this fact here (which suggestions looking at Milnors "Two Complexes which are homeomorphic but combinatorially distinct"), but I can't readily understand this.
Maybe there exists a simpler description that someone knows about?