In this paper, we propose a generic method for computing the rate-distortion-perception function (RDPF) of a multivariate Gaussian source under tensorizable distortion and perception metrics. Through the assumption of a jointly Gaussian reconstruction, we establish that the optimal solution of the RDPF belongs to the vector space spanned by the eigenvector of the source covariance matrix. Consequently, the multivariate optimization problem can be expressed as a function of the scalar Gaussian RDPFs of the source marginals, constrained by global distortion and perception levels. Utilizing this result, we devise an alternating minimization scheme based on the block nonlinear Gauss-Seidel method. This scheme solves optimally the optimization problem while identifying the optimal stagewise distortion and perception levels. Furthermore, the associated algorithmic embodiment is provided, along with the convergence and the rate of convergence characterization. Lastly, in the regime of "perfect realism", we provide the analytical solution for the multivariate Gaussian RDPF. We corroborate our findings with numerical simulations and draw connections to existing results.