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.

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.

One relation you do have: the product of the eigenvalues of $AB$ (counted by algebraic multiplicity) is the determinant of $AB$, and its absolute value is the product of the singular values of $AB$.