This question arose in the answers to Asaf Karagila's question Does ZFC prove that the universe is linearly orderable?. The answer there was that one can have a ZFC model with no global linear ordering of the universe (and consequently also no global well-ordering). The question here is whether we can separate these two principles from each other.
Question. Is it consistent with ZFC that there is a global linear ordering of the universe but no global well-ordering of the universe?
More specifically, there are two forms of the question, depending on whether one requires the classes to be definable classes as in ZFC, or whether one allows classes in the sense of Gödel-Bernays set theory.
If ZFC is consistent, is there a ZFC model with a definable linear ordering of the universe, but no definable (with parameters) well-ordering of the universe?
If ZFC is consistent, is there a model of GB+AC with a class linear ordering of the universe, but no class well-ordering of the universe?
The answer to the other question showed that there can be models of ZFC having no definable linear ordering of the universe, because one can make a class forcing extension which adds generic sets in a homogeneous manner, which prevents any definition from ordering them. Can we somehow modify the construction to allow a linear order, but no well-order? I suspect that we can, but I also suspect it will be easier to do this with GB classes than to make them definable.