Can there be a global linear ordering of the universe without a global well-ordering of the universe? 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. 
 A: I would like to record here my observation that one natural way of
proceeding does not actually succeed.
Specifically, there is a very natural class forcing $\mathbb{P}$
to add a generic linear ordering of the universe, without adding
any sets, as follows: conditions in $\mathbb{P}$ are simply
set-sized linear orderings of any piece of the universe. We order
these conditions by $p\leq q\iff p$ includes $q$ as a suborder.
This forcing is $\kappa$-closed for any $\kappa$ and hence adds no
new sets. Meanwhile, it is dense for any set to be added to the
field of the order, and so the generic filter $G$ provides a
generic linear ordering of the universe. Basically, this forcing builds up a linear ordering of the universe by fitting each set generically into the order.
This forcing notion is exactly analogous to the forcing
$\mathbb{Q}$ of global AC, where one uses conditions that are
well-orderings of any part of the universe, ordered by
end-extension. That forcing also adds no new sets, and the generic
filter is a global well-ordering. Thus, every model of ZFC can be
extended to a model of GBC = GB + global AC without adding any
sets. This is why GBC is conservative over ZFC for first order
assertions.
Meanwhile, my observation is that adding the global linear order
via $\mathbb{P}$ will also create a global well-order. To see
this, suppose that $G\subset\mathbb{P}$ is $V$-generic. For any
condition $p$, a set-sized linear order, there is some large
$\kappa$ such that the field of $p$ does not mention any unbounded
subset of $\kappa$. Let $\lhd$ be a well-ordering of the unbounded
subsets of $\kappa$, and let $q$ extend $p$ by placing a copy of
$\lhd$ above the field of $p$. Thus, $q$ has as a suborder a
well-ordering of all the unbounded subsets of $\kappa$. So by
density, the generic linear ordering given by $G$ must also have,
for arbitrarily large $\kappa$, a suborder that is a well-ordering
of all of the unbounded subsets of $\kappa$. This is enough to
define a global well-ordering of the universe, since we can say
that $X$ preceeds $Y$ if the transitive closure of $X$ has smaller
cardinality than $Y$, or they have the same size transitive
closure, but for the smallest $\kappa$ having in $G$ a
well-ordering of the unbounded subsets of $\kappa$, such that $\kappa$ is large enough to code these sets, that $X$ is
coded by an earlier such set than $Y$ is.
So the naive attempt to add a global linear order without adding a
global well-ordering doesn't work.
