We report on a search for a gravitationally bound solar axion halo using data from the Global Network of Optical Magnetometers for Exotic physics searches (GNOME), a worldwide array of magnetically shielded atomic magnetometers with sensitivity to exotic spin couplings. Motivated by recent theoretical work suggesting that self-interacting ultralight axions can be captured by the Sun's gravitational field and thermalize into the ground state, we develop a signal model for the pseudo-magnetic fields generated by axion-proton gradient couplings in such a halo. The analysis focuses on the fifth GNOME Science Run (69 days, 12 stations), employing a cross-correlation pipeline with time-shifted daily modulation templates to search for the global, direction-dependent, monochromatic signal expected from a solar axion halo. No statistically significant candidate signals are observed. We set 95% confidence-level upper limits on the amplitude of the axion-induced pseudo-magnetic field over the frequency range
≈0.05−20 Hz, translating to constraints on the linear and quadratic axion-proton couplings for halo densities predicted by gravitational capture models and for the maximum overdensities allowed by planetary ephemerides. In the quadratic coupling case, our limits surpass existing astrophysical bounds by over two orders of magnitude across much of the accessible parameter space.