Complement for joint POVMs?
I've come to the conclusion that the definition of the complement as I've written doesn't make sense for POVMs. It assumes that the product of two elements is a simultaneous event which is false. 

What I found is, because I'm dealing with probabilities and for me this was a "fail" event, I just switch my notion to 1 - "success" and I don't have to use any complements.