The following question is similar to that one, but it adds some features not included in that question.
Working in $\sf ZF+ Classes$, if we axiomatize the existence of a well order on the class of all cardinals that extends the usual $\leqslant$ relation on cardinals, would that enact any form of Choice? And what is that form?
$\sf ZF + Classes$, a conservative extension of $\sf ZF$, adds another sort (upper case representing classes) to the original language of $\sf ZF$ (lower case standing for sets); write all axioms of $\sf ZF-Ext$ in lower case; axiomatize that classes are extensional; sets are classes; elements are sets, and free construction of classes after formulas in lower case. Formally:
Axioms
Sorting: $ \exists X : X=y$
Membership: $ X \in Y \to \exists z: X=z$
Extensionality: $\forall A \, \forall B: \forall x \, (x \in A \leftrightarrow x \in B) \to A=B $
Comprehension: $ \big {(}\exists X \, \forall y \, (y \in X \leftrightarrow \phi) \big{)}$; where $\phi$ written in lower case.
Set axioms: all axioms of $\sf ZF-Ext$ written in lower case.
Well ordering cardinals: $\exists R: R\text { is a well order on} \operatorname {Card} \land \leqslant \, \subseteq R $
Where $\operatorname {Card} $ is the class of all Scott cardinals; "$\leqslant$" is the usual order on cardinals.
A related question is the same one but about when we weaken the last axiom to linear ordering?
