This is mostly a reference request, as this must be well known!
Let $A$ and $B$ be two real symmetric matrices, one of which is positive definite. Then it is easy to see that the product $AB$ (or $BA$, which has the same eigenvalues) is similar to a symmetric matrix, so has real eigenvalues. Now take the vectors of eigenvalues of $A$ and of $B$, sorted in decreasing order, and let their componentwise product be $ab$. Then the question is: what is known about the relationship (for instance, inequalities) between $ab$ and the vector of eigenvalues of the product $AB$ (also taken in decreasing order). Some experimentation gives the conjecture that there is a majorization order between them, for instance. This must be well-known!