Under what conditions is the product of two invertible diagonalizable matrices diagonalizable?

The answer in this question gives an example for the statement

product of two invertible diagonalizable matrices is not diagonalizable.

My question is:

Are there some conditions, perhaps on the eigenvalues and eigenvectors of matrices, under which the product of two invertible diagonalizable matrices is diagonalizable?

If two matrices $\;A,B\;$ are simultaneously diagonalizable (and thus they commute, for example), we get $$\begin{cases}A=P^{-1}D_1P\\{}\\B=P^{-1}D_2P\end{cases}\;\;\;\implies AB=P^{-1}D_1D_2P$$

and clearly $\;D_1D_2\;$ is also diagonal.

If you want to diagonalize a matrix using unitary operators, then it must be normal, i.e.

$A=UDU^\dagger$ if and only if $AA^\dagger = A^\dagger A$.

For your question, you would require that the product is normal:

$AB(AB)^\dagger=(AB)^\dagger AB$,

i.e., $ABB^\dagger A^\dagger = B^\dagger A^\dagger A B$