We certainly can't predict phases: e.g. multiply $A$ by a complex number $\omega$ with $|\omega|=1$, and you don't change any of the singular values but you multiply the eigenvalues of $A$ and $AB$ by $\omega$.
Consider the case $A = I$, $B$ a unitary $n \times n$ matrix. Then the singular values of $A$, $AB$, $BA$ are all $1$, but the eigenvalues of $B$ could be any $n$ points on the unit circle.