 Rank-shifting automorphism construction of NFU

In a model of ZFC, choose an external automorphism $j$ and an ordinal $\alpha$ with $j(\alpha )>\alpha$. Set $D_i=V_{j^i(\alpha )}$ for integer $i$, and declare the sets of type $i+1$ to be its objects that are subsets of $D_i$. Ordinary membership between successive types satisfies typed comprehension and set-extensionality. The automorphism shifts all types, yielding NFU by the type-zero collapse. A Skolem hull of integer-indexed increasing ordinal indiscernibles supplies such a $j$ by shifting the indices. The model is externally ill-founded; a transitive well-founded universe cannot have such an automorphism.