In the BH-galaxy co-evolution framework, most of the star-formation (SF) and the black hole (BH) accretion is expected to take place in highly obscured conditions. Thus, obscured AGN are difficult to identify in optical or X-ray bands, but shine bright in the IR. Moreover, X-ray background (XRB) synthesis models predict that a large fraction of the yet-unresolved XRB is due to the most obscured (Compton thick, CT) of these AGN. In this work, we investigate the synergies between putative IR missions (using SPICA, proposed for ESA/M5 but withdrawn in October 2020, and Origins Space Telescope, OST, as `templates) and the X-ray mission Athena, which should fly in early 2030s, in detecting and characterizing AGN, with a particular focus on the most obscured ones. Using an XRB synthesis model, we estimated the number of AGN and the number of those which will be detected in the X-rays. For each AGN we associated an optical-to-FIR SED from observed AGN with both X-ray data and SED decomposition, and used these SEDs to check if the AGN will be detected by SPICA-like or OST at IR wavelengths. We expect that, with the deepest Athena and SPICA-like (or OST) surveys, we will be able to detect in the IR more than $90,%$ of all the AGN (down to L$_{2-10text{keV}} sim 10^{42},$erg/s and up to $z sim 10$) predicted by XRB synthesis modeling, and we will detect at least half of them in the X-rays. Athena will be extremely powerful in detecting and discerning moderate- and high-luminosity AGN. We find that the most obscured and elusive CT-AGN will be exquisitely sampled by SPICA-like mission or OST and that Athena will allow a fine characterization of the most-luminous ones. This will provide a significant step forward in the process of placing stronger constraints on the yet-unresolved XRB and investigating the BH accretion rate evolution up to very high redshift ($z ge 4$).
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