What is a non-logical concrete example of a choiceless elementary topos? What is a non-logical concrete example of a choiceless elementary topos?
The elementary topoi made from choiceless models of ZF, or stuff like the realizability topos, are (I believe) choiceless, but is there a concrete example from other branches of mathematics?

Consider the topos corresponding to the topological space $\mathbb{C}$, with the object $\mathfrak{S}$ being the sheaf of analytic functions.  Then the differentiation operator $D : \mathfrak{S} \to \mathfrak{S}$ is an epimorphism (essentially because $\mathbb{C}$ is locally simply connected) - so the interpretation of the formula $\forall f \in \mathfrak{S}, \exists F \in \mathfrak{S}, D(F) = f$ is $\top$.  However, because any nonempty open subset of $\mathbb{C}$ contains a punctured open disc, $D$ does not have a right inverse when restricted to any nonempty open subset; therefore, the interpretation of the formula $\exists I \in \mathfrak{S}^{\mathfrak{S}}, D \circ I = \operatorname{id}_{\mathfrak{S}}$ is $\bot$.
