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We compute the spectral statistics of the sum H of two independent complexWishart matrices, each of which is correlated with a different covariancematrix. Random matrix theory enjoys many applications including sums andproducts of random matrices. Typically ensembles with correlations among thematrix elements are much more difficult to solve. Using a combination ofsupersymmetry, superbosonisation and bi-orthogonal functions we are able todetermine all spectral k-point density correlation functions of H for arbitrarymatrix size N. In the half-degenerate case, when one of the covariance matricesis proportional to the identity, the recent results by Kumar for the jointeigenvalue distribution of H serve as our starting point. In this case theensemble has a bi-orthogonal structure and we explicitly determine its kernel,providing its exact solution for finite N. The kernel follows from computingthe expectation value of a single characteristic polynomial. In the generalnon-degenerate case the generating function for the k-point resolvent isdetermined from a supersymmetric evaluation of the expectation value of kratios of characteristic polynomials. Numerical simulations illustrate ourfindings for the spectral density at finite N and we also give indications howto do the asymptotic large-N analysis.

Spectral correlation functions of the sum of two independent complex Wishart matrices with unequal covariances