This question is inspired by the excellent question by Douglas Ulrich When is $L$-Rank definable in inner models of $V=L$?
Suppose $M \in L$ is a countable model of $ZFC$, and furthermore suppose $M \vDash V \neq L$. These models have the funny property that, although every set in them is constructible, the model does not recognize this fact -- the "nonconstructible" sets simply arise at a level of the $L$-hierarchy that is greater than the ordinals of $M$. Provided any countable $GBC$ ZFC$ model exists, then such models abound -- for example, we can easily build forcing extensions of countable models by building an $M$-generic filter directly, and the forcing extension will not recognize that the generic is constructible.
In the answer to the question linked above it is shown that such a model $M$ cannot define the $L$-order on all of its members, at least not with the usual definition -- it can only define $<_L$ for those elements that it recognizes as constructible (that is, for members of $L^M$). However, I am interested whether this limitation is inherent, or simply a limitation of definability over $M$. In particular, is it consistent for us to add the $L$-order on all of $M$, as a class, without destroying $ZFC$? I will phrase the question in terms of $GBC$ models, to make the use of classes explicit.
Is it consistent that $M \in L$ is a countable model of $GBC$ with $M \vDash V \neq L$, and there is class $U \in M$ such that $U$ gives the $L$-order on $M$? (That is, $\langle x,y \rangle \in U$ if and only if $x <_L y$, for all sets $x, y \in M$).
Note: In the context of $GBC$ the statement $V \neq L$ may be ambiguous -- I intend it to refer only to sets, and not to classes, e.g. $V \neq L$ means "there exists a nonconstructible set."
Such a $U$ would give us a proper class well order with order type larger than $ORD^M$, but this does not seem immediately problematic - many such well orders are definable over any model of $ZFC$. One way in which such a $U$ might be inconsistent is if, from $U$, we could show how to "construct" every set in $M$, but these constructions would be of more-than-$ORD^M$ length, and so this might not directly contradict $V \neq L$.
If such a $U$ is inconsistent, it would be nice to see why this limitation exists. On the other hand, if we can have such a $U$, is it universally possible?
If such a $U$ is consistent, are there any restrictions on the models $M$ that can have them?
For example, it might be consistent only if every member of $M$ has $L$-rank "not too much greater" than $ORD^M$.
EDIT: changed a stray occurence of "GBC" to "ZFC" in paragraph 1, for clarity.
