In the 1990s, Blackadar and Kirchberg introduced NF algebras, which arise as limits of inductive systems where the connecting *-homomorphisms are replaced with asymptotically multiplicative completely positive contractive maps. It turns out, asymptotic multiplicativity is enough to guarantee a multiplicative structure on the inductive limit– meaning the limit is a C*-algebra (in fact, a nuclear quasidiagonal C*-algebra). In this talk, I will describe a generalization of Blackadar and Kirchberg’s inductive limits, which further relaxes asymptotic multiplicativity to asymptotic orthogonality preserving. Though the limit is no longer an algebra, the rich structure of orthogonality preserving (order zero) maps allows us to nonetheless detect a nuclear C*-structure on the inductive limit. This is joint work with Wilhelm Winter.