An internal category $\mathbb A$ in an elementary topos $\mathcal E$ is called an (intrinsic) stack if the indexed category that it represents, $(X\in \mathcal E) \mapsto \mathcal{E}(X,\mathbb A) \in \rm Cat$, is a stack for the regular topology of $\mathcal E$. An internal functor is called a weak equivalence if it is internally essentially surjective and fully faithful; this is strictly weaker than being an internal equivalence if the axiom of choice does not hold in $\mathcal{E}$. However, if $f:\mathbb A \to \mathbb B$ is a weak equivalence and $\mathbb C$ is a stack, then the induced functor $\mathrm{Cat}(\mathcal E)(\mathbb B,\mathbb C)\to \mathrm{Cat}(\mathcal E)(\mathbb A,\mathbb C)$ is an equivalence.
Say that $\mathcal E$ satisfies the axiom of stack completions (ASC) if every internal category admits a weak equivalence to an internal category that is a stack (see Bunge and Hermida, "Pseudomonadicity and 2-stack completions"). If $\mathcal E$ is a Grothendieck topos, then stack completions can be constructed using the small object argument, as fibrant replacements in the model structure on $\mathrm{Cat}(\mathcal E)$ constructed by Joyal and Tierney ("Strong stacks and classifying spaces"), so it satisfies ASC.
My question is whether realizability toposes also satisfy ASC?
I suspect the answer is no, because realizability toposes can contain small categories that are weakly complete, meaning that their stack completions are indexed-complete, but the last I heard it was unknown whether they can contain small categories that are strongly complete in the internal-category sense. If they satisfied ASC then this distinction would not be an issue since stack completion could be applied to the original weakly complete internal category to produce a strongly complete one. However, if this is the case, I would like to see a concrete example or proof that ASC does not hold in a realizability topos.
