Localized electron spins can couple magnetically via the Ruderman–Kittel–Kasuya–Yosida interaction even if their wave functions lack direct overlap. Theory predicts that spin–orbit scattering leads to a Dzyaloshinskii–Moriya type enhancement of this indirect exchange interaction, giving rise to chiral exchange terms. Here we present a combined spin-polarized scanning tunneling microscopy, angle-resolved photoemission, and density functional theory study of MnO_2 chains on Ir(100). Whereas we find antiferromagnetic Mn–Mn coupling along the chain, the inter-chain coupling across the non-magnetic Ir substrate turns out to be chiral with a 120° rotation between adjacent MnO_2 chains. Calculations reveal that the Dzyaloshinskii–Moriya interaction results in spin spirals with a periodicity in agreement with experiment. Our findings confirm the existence of indirect chiral magnetic exchange, potentially giving rise to exotic phenomena, such as chiral spin-liquid states in spin ice systems or the emergence of new quasiparticles