We examine the impact of interfacial phonons on the superconducting state of FeSe/SrTiO3 developing a materials' specific multiband, full bandwidth, anisotropic Eliashberg theory for this system. Our selfconsistent calculations highlight the importance of the interfacial electron-phonon interaction, which is hidden behind the seemingly weak coupling constant λm=0.4, in mediating the high-Tc, and explain other puzzling experimental observations like the s-wave symmetry and replica bands. We discover that the formation of replica bands has a Tc decreasing effect that is nevertheless compensated by deep Fermi-sea Cooper pairing which has a Tc enhancing effect. We predict a strong coupling dip-hump signature in the tunneling spectra due to the interfacial coupling.Superconductivity in monolayer-thick FeSe on SrTiO 3 reaches amazingly high transition temperatures of typically T c =50-70 K [1-6] and up to 100 K [7], much higher than the 8-K value of bulk FeSe [8]. A coupling between SrTiO 3 phonons and FeSe electrons occurs at the FeSe/SrTiO 3 interface, which manifests itself as electron replica bands [6]. The value of this coupling is estimated by experiments to be around 0.4, thus it is commonly believed to moderately enhance T c but not be enough to explain it [6,9,10].The superconducting state in iron-based superconductors is customarily associated with residual spin fluctuations due to the remnant quasi-nesting between electron and hole Fermi sheets that give rise to a sign alternating gap [11]. However, for FeSe/STO the situation is markedly different. Charge transfer at the interface induces electron doping in FeSe [3,4], leading to a distinct Fermi surface consisting of only two electron sheets around the corners of the tetragonal Brillouin zone (M point) [12]. The observed anisotropic superconducting gap has a more conventional form with plain s-wave symmetry [13] and is thus nodeless in the entire Brillouin zone. Furthermore, angular resolved photoemission spectroscopy (ARPES) measurements [6] reveal an interfaceinduced electron-phonon interaction (EPI) between FeSe electrons and polar STO phonons that is strongly peaked at the q=0 phonon wavevector [6,[14][15][16][17][18]. There is growing experimental evidence for the pivotal role of such interfacial phonons in engineering high-T c heterostructures that involve FeSe [5,9,19] or even FeAs [20] monolayers.Although ab initio calculations confirm the existence of small-q phonons as a strictly interfacial phenomenon in FeSe/STO [12,[21][22][23] and indicate the importance of the coupling between substrate phonons and FeSe electrons [24], the estimated low value of the electron-phonon coupling constant (λ ≤ 0.4) has been widely considered insufficient to explain the impressive T c enhancement unless another, dominant pairing mechanism is at play [6]. On the other hand, Eliashberg calculations within a single band model suggest that interfacial phonons may lead to the high T c with a coupling of merely half of that estimated by experiments [25]. However, a mater...