Suppose you have two i.i.d $K \times N$ random matrices $A$ and $B$. Both matrices have complex valued entries taken from a Gaussian distribution with $\mu = 0$ and normalised variance $\sigma^2 = \frac{1}{N}$. What distribution would the matrix $A(B^{*})$ have, where $\{\cdot\}^{*}$ denotes the conjugate transpose (hermitian) of the matrix $\{\cdot\}$?
In particular I am interested to know whether the asymptotic eigenvalue distribution can be given in a closed form for the self-adjoint matrix:
$$A(B^{*})+B(A^{*}).$$