We study the two-point correlation functions for the zeroes of systems of SO(n+1)-invariant Gaussian random polynomials on RPn and systems of Isom(Rn) -invariant Gaussian analytic functions. Our result reflects the same “repelling”, “neutral”, and “attracting” short-distance asymptotic behavior, depending on the dimension, as was discovered in the complex case by Bleher, Shiffman, and Zelditch. We then prove that the correlation function for the Isom(Rn)-invariant Gaussian analytic functions is “universal”, describing the scaling limit of the correlation function for the restriction of systems of the SO(k+1)-invariant Gaussian random polynomials to any n-dimensional C2 submanifold M⊂RPk. This provides a real counterpart to the universality results that were proved in the complex case by Bleher, Shiffman, and Zelditch.