  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.
