  Due to a very low production rate of electron anti-neutrinos ($\bar{\nu}_e$)
via nuclear fusion in the Sun, a flux of solar $\bar{\nu}_e$ is unexpected. An
appearance of $\bar{\nu}_e$ in solar neutrino flux opens a new window for the
new physics beyond the standard model. In particular, a spin-flavor precession
process is expected to convert an electron neutrino into an electron
anti-neutrino (${\nu_e\to\bar{\nu}_e}$) when neutrino has a finite magnetic
moment. In this work, we have searched for solar $\bar{\nu}_e$ in the
Super-Kamiokande experiment, using neutron tagging to identify their inverse
beta decay signature. We identified 78 $\bar{\nu}_e$ candidates for neutrino
energies of 9.3 to 17.3 MeV in 2970.1 live days with a fiducial volume of 22.5
kiloton water (183.0 kton$\cdot$year exposure). The energy spectrum has been
consistent with background predictions and we thus derived a 90% confidence
level upper limit of ${4.7\times10^{-4}}$ on the $\nu_e\to\bar{\nu}_e$
conversion probability in the Sun. We used this result to evaluate the
sensitivity of future experiments, notably the Super-Kamiokande Gadolinium
(SK-Gd) upgrade.
