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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 field oscillating at an (as of yet) unknown frequency. The frequency depends linearly on the ALP mass and plausibly ranges from 10⁻²²to 10 eV/c². This motivates broadband search approaches. We report on a direct search for ALP dark matter with an interferometer composed of two atomic K-Rb-³He comagnetometers, one situated in Mainz, Germany, and the other in Kraków, Poland. We leverage the anticipated spatio-temporal coherence properties of the ALP field and probe all ALP-gradient-spin interactions covering a mass range of nine orders of magnitude. No significant evidence of an ALP signal is found. We thus place new upper limits on the ALP-neutron, ALP-proton and ALP-electron couplings reaching belowg_aNN < 10⁻⁹ GeV⁻¹,g_aPP < 10⁻⁷ GeV⁻¹andg_aee < 10⁻⁶ GeV⁻¹, respectively. These limits improve upon previous laboratory constraints for neutron and proton couplings by up to three orders of magnitude.