The question is if there is a combination/s of tilts/angles, for all three tetrahedrons, where any of the combinations between the vertices, which are not part of the 3-4-5 triangle, give an equilateral triangle ?
The axes of tilt are the edges of the 3-4-5 triangle.
In the depicted co-planar case, two sides of the resulting triangle appear to be the same, and very close in value to the third.