I think that is right, Helios. The dihedral angle of a regular tet is about 70.5 degrees,
Suppose I allow two kinds of tet. Can it be done? Please tell us if you know.
[This may not be absolutely on topic, because all we need to accomplish what Etera is talking about is some sort of tetrahedral triangulation of space, which I'm pretty sure exists (if we relax the regularity condition slightly). But it's not a bad exercise for the imagination to think about it. Helios might be a good teacher here.]