My idea was to try to use paper efficiently and equally for all faces and edges. Instead of having every face join at the edge, I opted to have all but one edge not touching. After some trial and error I found a simple lock from where two of the edges come together, as well as the reference for a 15 degree angle (which I have no doubt has been found before).
I'm happy with the result, despite the fact that the lock ruins the otherwise clean faces of the model. There is way to get the excess paper from the sides of the locking face to hold a lock behind it, but it doesn't hold as tightly. Let's just say the colour change on the lock means the tetrahedron could double as a snow-capped mountain or Christmas tree.