There is a general belief that the universe is totally ordered. What does it mean for things in the universe to no longer be totally ordered?
Well, it is possible to assign a total ordering to "things" in the universe, provided we can unambiguously distinguish these things. All one needs to do is satisfy the total ordering conditions: antisymmetry, transitivity, and the connex property.
I think the core of your question instead is whether there are total orderings of the universe that are useful to us, possibly to explain physical phenomena, and the answer to that is that physicists tend to view the universe more in terms of symmetries rather than orderings, and so research into orderings of the universe is not as advanced as other areas of physics. (In particular, I can't seem to find any peer-reviewed articles on the topic.)