A $\beta$-model of a set theory (or higher-order-arithmetic theory) is a model $M$ of that theory which is correct about well-foundedness: if $x\in M$ is an illfounded relation, then there is some $a\in M$ which is a subset of the domain of $x$ with no minimal element.
I know a bit about $\beta$-models of ZFC- and $Z_2$-like theories - in particular, $\beta$-models of ZFC are just well-founded models of ZFC - but I've realized that embarrassingly I know nothing whatsoever about $\beta$-models of NF-like theories. The existence of a "reasonable" set theory without $\beta$-models would be amazing, so I'm sure that $(i)$ there are easy ways to construct $\beta$-models of NFU (or even strengthenings like NFU + Infinity + Choice) and $(ii)$ there are no major reasons to be more skeptical of the existence of a $\beta$-model of NF than of the mere consistency of NF.
That said, I still don't see how to whip them up. So:
How does one construct a $\beta$-model of NFU?
I'm especially interested in $\beta$-models of strong extensions of NFU (like NFU + Choice + Infinity + "Cantorian sets"). I'm also interested in heuristic arguments about why (I assume!) the $\beta$-consistency of NF should be equiplausible with the consistency of NF.