Multiplicative non-Hermitian perturbations of classical $\beta$-ensembles

We compute the joint eigenvalue distribution for a multiplicative non-Hermitian perturbation $(I+i\mathrm{\Gamma })H$, $\mathrm{rank}\mathrm{\Gamma }=1$ of a random matrix $H$ from the Gaussian, Laguerre, and chiral Gaussian $\beta$-ensembles.