Live coded on the 24th of June 2016 in 95 minutes. Needs support for transform-style: preserve-3d (so no IE, not even 11). Tested and works in Firefox 47, Chrome 51, Opera 39 on Windows 7. Should probably work in Edge and Safari too, but I cannot test. If you can test, please let me know how it works.
Update: clipping triangles to themselves seems not to work in Safari (tweet, tweet). Works in Edge except for thickness of edges as Edge doesn't support vector-effect (tweet, tweet).