This is consistent. Kanovei constructed a model $M$ with an infinite Dedekind finite set of reals which is lightface projectively definable (see "On the Nonemptiness of Classes in Axiomatic Set Theory"). By descending to $L(R),$ we can further assume it satisfies $V=L(R).$

Clearly choice fails in this model. Since it satisfies $V=L(R),$ every set is definable from an ordinal and a real. Of course, any ordinal is definable from an $\aleph,$ so we just need to check that every real is cardinal definable.

Let $A$ be the definable Dedekind finite set of reals, and fix a canonical surjection $f: A \rightarrow \omega$ (as exists for any infinite set of reals). Let $\langle q_n \rangle$ be an enumeration of the rationals. Define $A_r=f^{-1}(\{n: q_n<r\}).$ For $r<s,$ $|A_r|<|A_s|$ since these are Dedekind finite sets. Clearly $r$ is definable from the cardinality of $A_r,$ so we're done.

This is consistent. Kanovei constructed a model $M$ with an infinite Dedekind finite set of reals which is lightface projectively definable. By descending to $L(R),$ we can further assume it satisfies $V=L(R).$

Clearly choice fails in this model. Since it satisfies $V=L(R),$ every set is definable from an ordinal and a real. Of course, any ordinal is definable from an $\aleph,$ so we just need to check that every real is cardinal definable.

Let $A$ be the definable Dedekind finite set of reals, and fix a canonical surjection $f: A \rightarrow \omega$ (as exists for any infinite set of reals). Let $\langle q_n \rangle$ be an enumeration of the rationals. Define $A_r=f^{-1}(\{n: q_n<r\}).$ For $r<s,$ $|A_r|<|A_s|$ since these are Dedekind finite sets. Clearly $r$ is definable from the cardinality of $A_r,$ so we're done.

Reference:

Kanovej, V. G., On the nonemptiness of classes in axiomatic set theory, Math. USSR, Izv. 12, 507-535 (1978). ZBL0427.03044.