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Geological and geochemical evidence can be interpreted as indicating strong hysteresis in global climate during the Neoproterozoic glacial events (similar to 630 Ma and similar to 715 Ma). Standard climate theory only allows such strong hysteresis if global climate enters a fully-glaciated "Snowball" state. However, the survival of photosynthetic, eukaryotic, marine species through these glaciations may indicate that there were large areas of open ocean. A previously-proposed "Slushball" model for Neoproterozoic glaciations could easily explain the survival of these organisms because it has open ocean throughout the tropics, but there is only a small amount of hysteresis associated with the Slushball state. In this paper a new state of global climate, the "Jormungand" state, is proposed. In this state the ocean is very nearly globally ice-covered, but a very small strip of the tropical ocean remains ice-free. The low ice latitude of the Jormungand state is a consequence of the low albedo of snow-free (bare) sea ice. If the ice latitude propagates into the subtropical desert zone, it can stabilize without collapsing to the equator because subtropical ice-covered regions have a relatively low top-of-atmosphere albedo as a result of the exposure of bare sea ice and relatively lower cloud cover. Moreover, there is strong hysteresis associated with the Jormungand state as greenhouse gas levels are varied because of the high albedo contrast between regions of bare and snow covered sea ice. The Jormungand state is illustrated here in two different atmospheric global climate models and in the Budyko-Sellers model. By offering a scenario that could explain both strong hysteresis in global climate and the survival of life, the Jormungand state represents a potential model for Neoproterozoic glaciations, although further study of this issue is needed.