Sodium-conducting sulfide glasses are promising materials for the next generation of solid-state batteries. Deep insight into the glass structure is required to ensure a functional design and tailoring of vitreous alloys for energy applications. Using pulsed neutron diffraction supported by first-principles molecular dynamics, we show a structural diversity of Na2S-As2S3 sodium thioarsenate glasses, consisting of long corner-sharing (CS) pyramidal chains CS-(AsSS2/2)(k), small AspSq rings (p + q <= 11), mixed corner- and edge-sharing oligomers, edge-sharing (ES) dimers ES-As2S4, and isolated (ISO) pyramids ISOAsS3, entirely or partially connected by sodium species. Polysulfide S-S bridges and structural units with homopolar As-As bonds complete the glass structure, which is basically different from structural motifs predicted by the equilibrium phase diagram. In contrast to superionic silver and sodium sulfide glasses, characterized by a significant population of isolated sulfur species S-iso (0.20 < S-iso/S-tot < 0.28), that is, sulfur connected to only mobile cations M+ with a usual M/S-iso stoichiometry of 2, poorly conducting Na2S-As2S3 alloys exhibit a modest S-iso fraction of 6.2%.