Coupled system instability
Assuming both black boxes are LTI systems, that's reasonably easy to do.
Let's call the transfer function of the first box $H_A(\omega)$ and the second one $H_B(\omega)$. The open look transfer function is simply the product of the, i.e. $H_O(\omega) = H_A(\omega) \cdot H_B(\omega)$
A sufficient stability condition is $|H_O(\omega)| &lt; 1$. For an actual acoustic feedback loop that's really all there is to it since there is always plenty of group delay in the system and the phase is spinning so fast that any type of phase criteria is pointless.
So measure the transfer functions (fully calibrated !), multiply them and check whether the max gain is smaller than unity.