Abstract: We investigate the far-infrared (far-IR) incidence of X-ray-selected active galactic nuclei (AGNs) and galaxies that do not host an AGN (non-AGNs) as a function of the stellar mass (M_), star formation rate (SFR), and specific black hole accretion rate (?sBHAR), using data from five well-characterized extragalactic fields (COSMOS, XMM-LSS, Stripe82, ELAIS-S1, and CDFS-SWIRE). We constructed spectral energy distributions (SEDs) using optical-to-far-IR photometry to derive host galaxy properties and assess AGN obscuration, while X-ray absorption was quantified using the 4XMM-DR11s catalogue. Our final sample comprises 172 697 non- AGN galaxies (53% Herschel-detected) and 2417 X-ray AGNs (73% Herschel-detected), with 10 < log [M*=M?] < 12 and 0 < z < 2. We find that X-ray AGNs exhibit a relatively flat far-IR detection rate across stellar mass and specific SFR (sSFR = SFR=M*), unlike non-AGN galaxies, where detection correlates strongly with star formation. Far-IR detection among AGNs decreases with increasing ?sBHAR, even as their SFR rises. Our results suggest that X-ray AGNs are preferentially found in gas-rich environments, where both star formation and black hole accretion are fuelled by the presence of cold gas. The far-IR incidence of X-ray AGNs remains high across all sSFR bins, indicating that these AGNs can coexist with ongoing star formation for extended periods, in line with a scenario in which AGNs feedback regulates rather than abruptly quenches star formation. We also find that comparing AGNs and non-AGN SFRs without separating Herschel-detected from non-detected sources introduces biases. Obscured AGNs show 10% higher far-IR detection rates than unobscured ones, yet at similar ?sBHAR, unobscured AGNs tend to have higher SFR. This may indicate that obscured AGNs reside in dustier environments where moderate star formation still contributes to far-IR emission. Our results support a scenario in which AGNs and star formation coexist in gas-rich galaxies, with AGNs feedback acting as a regulatory process over extended timescales and not necessarily quenching.

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