We will introduce Nairian Models, which are models of the Axiom of Determinacy that have non-trivial structure above their Theta. We will then show how to force the Axiom of Choice over these models. Then we will explain why the square principle fails in the aforementioned generic extensions of Nairian Models, and state some other combinatorial principles that hold in these extension of Nairian Models.Parts of this talk are joint with Douglas Blue and Paul Larson.