Consider the embedding of a W*- (respectively C*-)algebra into its own ultrapower. For particularly nice objects (for example the hyperfinite II_1 factor or the Jiang-Su algebra), these embeddings are always unitarily conjugate to the natural inclusion map. In this talk, we discuss the related notion of when a II_1 factor admits embeddings into an ultrapower with factorial relative commutant. This is a strengthening of the McDuff property which we call the "uniform super McDuff property"; using model-theoretic techniques, we show that this property is generic in the class of II_1 factors. This is based on joint work with Isaac Goldbring, David Jekel, and Srivatsav Kunnawalkam Elayavalli.