Axion-like particles (ALPs) arise from well-motivated extensions to the Standard Model and could account for dark matter. ALP dark matter would manifest as a nearly monochromatic field oscillating at an (as of yet) unknown frequency. The frequency depends on the ALP mass, which could plausibly range from $10^{-22}$ eV/$c^2$ to $10$ eV/$c^2$. We report on a direct search for ALP dark matter through the ALP-nucleon interaction by interfering the signals of two atomic K-Rb-$^3$He comagnetometers, with one situated in Mainz, Germany, and the other in Krak\'ow, Poland. We use the ALP dark matter's spatiotemporal coherence properties assuming the standard halo model of dark matter in the Milky Way to improve the sensitivity and exclude spurious candidates. The search extends over nine orders of magnitude in ALP mass. In this range, no significant evidence of an ALP signal is found. We thus place new upper limits on the ALP-neutron and ALP-proton couplings of $g_{aNN}<10^{-5}$ GeV$^{-1}$ and $g_{aPP}<5 \times 10^{-4}$ GeV$^{-1}$ at a mass of $10^{-22}$ eV/$c^2$ and extending to a mass of $10^{-15}$ eV/$c^2$ where the upper limits reach below $g_{aNN}<10^{-9}$ GeV$^{-1}$ and $g_{aPP}<10^{-7}$ GeV$^{-1}$, respectively. For both neutron and proton couplings, this work is an improvement of up to four orders of magnitude compared to previous laboratory constraints.
