Hydride molecules lie at the base of interstellar chemistry, but the synthesis of sulfuretted hydrides is poorly understood. Motivated by new observations of the Orion Bar PDR - 1 resolution ALMA images of SH+; IRAM 30m detections of H2S, H2S34, and H2S33; H3S+ (upper limits); and SOFIA observations of SH - we perform a systematic study of the chemistry of S-bearing hydrides. We determine their column densities using coupled excitation, radiative transfer as well as chemical formation and destruction models. We revise some of the key gas-phase reactions that lead to their chemical synthesis. This includes ab initio quantum calculations of the vibrational-state-dependent reactions SH+ + H2 <-> H2S+ + H and S + H2 <-> SH + H. We find that reactions of UV-pumped H2 (v>1) with S+ explain the presence of SH+ in a high thermal-pressure gas component, P_th~10^8 cm^-3 K, close to the H2 dissociation front. However, subsequent hydrogen abstraction reactions of SH+, H2S+, and S with vibrationally excited H2, fail to ultimately explain the observed H2S column density (~2.5x10^14 cm^-2, with an ortho-to-para ratio of 2.9+/-0.3). To overcome these bottlenecks, we build PDR models that include a simple network of grain surface reactions leading to the formation of solid H2S (s-H2S). The higher adsorption binding energies of S and SH suggested by recent studies imply that S atoms adsorb on grains (and form s-H2S) at warmer dust temperatures and closer to the UV-illuminated edges of molecular clouds. Photodesorption and, to a lesser extent, chemical desorption, produce roughly the same H2S column density (a few 10^14 cm-^2) and abundance peak (a few 10^-8) nearly independently of n_H and G_0. This agrees with the observed H2S column density in the Orion Bar as well as at the edges of dark clouds without invoking substantial depletion of elemental sulfur abundances.
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