We discuss correlations between the neutrino-mass-embedded Standard Model CP phase d and the phases that originate from new physics which cause neutrino-sector unitarity violation (UV) at low energies. This study aims to provide one of the building blocks for machinery to diagnose non-unitarity, our ultimate goal. We extend the perturbation theory of neutrino oscillation in matter proposed by Denton et al. (DMP) to include the UV effect expressed by the alpha parametrization. By analyzing the DMP-UV perturbation theory to first order, we are able to draw a complete picture of the delta-UV phase correlations in the whole kinematical region covered by terrestrial neutrino experiments. Two regions exist with characteristically different patterns of the correlations: (i) the chiral-type [e(-i delta)alpha(mu e), e(-i delta)alpha(tau e), alpha(tau mu)] (Particle Data Group convention) correlation in the entire high-energy region vertical bar rho E vertical bar greater than or similar to 6 (g/cm(3)) GeV, and (ii) (blobs of the a parameters)-e(+/- i delta) correlation anywhere else. Some relevant aspects for the measurement of the UV parameters, such as the necessity of determining all the alpha(beta gamma) elements at once, are also pointed out.