Here is another proposal.
Let $T_{ZFC}$ = arithmetical truths that $ZFC$ "knows about", i.e., the set of arithmetic sentences $\phi$ such that $ZFC$ proves $\phi^\omega$, where $\phi^\omega$ is the set-theoretical statement that expresses "$\phi$ holds in the von Neumann interpretation".
$T_{ZFC}$ is an example of a natural r.e. theory whose overt axiomatization is not primitive recursive. Note that $T_{ZFC}$ is much stronger than $PA$, since includes all kinds of statements that are left undecided by $PA$, such as $Con(PA)$, $Con(PA + Con(PA)$, etc.
Moreover, one could argue that this theory is in some sense more natural than $PA$ since it is, implicitly, what mathematicians are really interested in.
Three notes:
There are axiomatizations of $T_{ZFC}$ that do not use the Craig trick; see this FOM posting for more information.
As observed by Kreisel, $T_{ZF}$ = $T_{ZFC + GCH}$ (the argument uses Gödel's constructible universe and absoluteness considerations).
In light of Harvey Friedman's programme of unearthing the deep role of large cardinals in our knowledge of the finite realm, one can also consider natural variants $T_{ZFC^{+}}$, where $ZFC^{+}$ is the result of augmenting $ZFC$ with appropriate large cardinals.