We have investigated the quadruple sulfur isotopic composition of inorganic sulfur-bearing phases from 13 carbonaceous chondrites of CM type. Our samples include 4 falls and 9 Antarctic finds. We extracted sulfur from sulfides, sulfates, and elemental sulfur (S0) from all samples. On average, we recover a bulk sulfur (S) content of 2.11 ± 0.39 wt.% S (1σ). The recovered sulfate, S0 and sulfide contents represent 25 ± 12%, 10 ± 7% and 65 ± 15% of the bulk S, respectively (all 1σ). There is no evidence for differences in the bulk S content between falls and finds, and there is no correlation between the S speciation and the extent of aqueous alteration. We report ranges of Δ33S and Δ36S values in CMs that are significantly larger than previously observed. The largest variations are exhibited by S0, with Δ33S values ranging between −0.104 ± 0.012‰ and +0.256 ± 0.018‰ (2σ). The Δ36S/33S ratios of S0 are on average −3.1 ± 1.0 (2σ). Two CMs show distinct Δ36S/33S ratios, of +1.3 ± 0.1 and +0.9 ± 0.1. We suggest that these mass independent S isotopic compositions record H2S photodissociation in the nebula. The varying Δ36S/Δ33S ratios are interpreted to reflect photodissociation that occurred at different UV wavelengths. The preservation of these isotopic features requires that the S-bearing phases were heterogeneously accreted to the CM parent body. Non-zero Δ33S values are also preserved in sulfide and sulfate, and are positively correlated with S0 values. This indicates a genetic relationship between the S-bearing phases: We argue that sulfates were produced by the direct oxidation of S0 (not sulfide) in the parent body. We describe two types of models that, although imperfect, can explain the major features of the CM S isotope compositions, and can be tested in future studies. Sulfide and S0 could both be condensates from the nebula, as the residue and product, respectively, of incomplete H2S photodissociation by UV light (wavelength <150 nm). This idea requires that FeS formation and the S0 condensation co-occur. As an alternative, ice accretion to the CM parent body could allow the delivery of S-MIF in CMs. In that case, sulfides would have been the only S-bearing condensate in CM precursors, and S0 would have been derived from the oxidation of H2S trapped in ices, after its photodissociation at low temperature (<500 K) in the nebula. In our models, the observations of H2S UV photodissociation is required to occur at the disk surface, and allowed in nebular environments with canonical C/O ratios. Vertical motions in the disk would redistribute phases that condensed at high altitude to the midplane, where they accreted in the phases that make up the chondritic matrix.