Context. Hydrogen sulfide (H2S) is thought to be efficiently formed on grain surfaces through the successive hydrogenation of sulfur atoms. Its non-detection so far in astronomical observations of icy dust mantles thus indicates that effective destruction pathways must play a significant role in its interstellar abundance. While chemical desorption has been shown to remove H2S very efficiently from the solid phase, in line with H2S gas-phase detections, possible ice chemistry triggered by the related HS radical have been largely disregarded so far, despite it being an essential intermediate in the H2S + H reaction scheme. Aims. We aim to thoroughly investigate the fate of H2S upon H-atom impact under molecular cloud conditions, providing a comprehensive analysis combined with detailed quantification of both the chemical desorption and ice chemistry that ensues. Methods. We performed experiments in an ultrahigh vacuum chamber at temperatures between 10 and 16 K in order to investigate the reactions between H2S molecules and H atoms on interstellar ice analogs. The changes in the solid phase during H-atom bombardment were monitored in situ by means of reflection absorption infrared spectroscopy (RAIRS), and desorbed species were complementarily measured with a quadrupole mass spectrometer (QMS). Results. We confirmed the formation of H2S2 via reactions involving H2S + H and quantified its formation cross section under the employed experimental conditions. Additionally, we directly assessed the chemical desorption of H2S by measuring the gas-phase desorption signals with the QMS, providing unambiguous desorption cross sections. Chemical desorption of H2S2 was not observed. The relative decrease of H2S ices by chemical desorption changed from ~85% to ~74% between temperatures of 10 and 16 K, while the decrease as the result of H2S2 formation was enhanced from ~15% to ~26%, suggesting an increasingly relevant sulfur chemistry induced by HS radicals at warmer environments. The astronomical implications are further discussed.