Is \theta_{13}^{PMNS} correlated with \theta_{23}^{PMNS} or not?

By postulating the relation \theta_{23} \simeq 45^\circ + \eta\theta_{13}, we seek preferable correction terms to tri-bi-maximal mixing and discuss their origins. Global analyses of the neutrino oscillation parameters favor \eta=\pm 1/\sqrt{2}; this corresponds to the relation found by Edy, Frampton, and Matsuzaki some years ago in the context of a T^\prime flavor symmetry. In contrast, the results of the \nu_\mu disappearance mode reported by the T2K and Super-Kamiokande collaborations seem to prefer \eta=0, which gives an almost maximal \theta_{23}. We derive a general condition for ensuring \theta_{23} \simeq 45^\circ + \eta\theta_{13} and find that the condition is complicated by the neutrino masses and CP violating phases. We investigate the condition under simplified environments and arrive at several correction terms to the mass matrices. It is found that the obtained correction terms can arise from flavor symmetries or one-loop radiative corrections.