the intersection of the tetrahedra

Why not formulate a convex optimization problem, or precisely a feasibility problem using the plane inequalities that you have? Let’s say, the two tetrahedra can be represented as A1.X + d1 <= 0 and A2.X + d2 <= 0 where the 4 rows of A1 and A2 store the a, b, c of four planes corresponding to the two tetrahedra in ax + by + cz + d <= 0, and column vectors d1 and d2 store the constants ie d. And also note that A1.X is matrix multiplication.

Represent (x, y, z) as the vector X.

Now basically you want to solve a feasibility problem for X like this:

minimize 0
subject to A1.X + d1 <= 0
           A2.X + d2 <= 0

Note that if the solver returns inf, that means there is no X which satisfies the above constraints. If the solver returns 0 (which is the value of the constant objective function), that means there is atleast one X which satisfies the constraints.

You can use cvxpy library for this. Here is a nice tutorial. Also cvxpy library goes well with numpy.

And I don’t think solving equations would work in this case as a tetrahedron is basically composed of four linear inequalities. So you have to solve inequalities in order to find a solution in their intersection region.