We investigate spectral properties of a Hermitised random matrix product which, contrary to previous product ensembles, allows for eigenvalues on the full real line. We find an explicit expression of the joint probability density function as a bi-orthogonal ensemble. As an interesting example, we focus on  the  product  of  GUE and LUE matrices and provide explicit expressions both for the bi-orthogonal functions and the correlation kernel.  Then a new  double-side kernel is found  at the origin, which is slightly different from the  Bessel kernel.