We investigate epistemic independence for choice functions in a multivariate setting. This work is a continuation of earlier work of one of the authors , and our results build on the characterization of choice functions in terms of sets of binary preferences recently established by De Bock and De Cooman . We obtain the independent natural extension in this framework. Given the generality of choice functions, our expression for the independent natural extension is the most general one we are aware of, and we show how it implies the independent natural extension for sets of desirable gambles, and therefore also for less informative imprecise-probabilistic models. Once this is in place, we compare this concept of epistemic independence to another independence concept for choice functions proposed by Seidenfeld , which De Bock and De Cooman  have called S-independence. We show that neither is more general than the other.