During my recent debate on Compatible Free Will as opposed to Libertarian Free Will, which was supposed to be about whether the Bible teaches Libertarian Free Will and ended up being about whether the word "choose" requires Libertarian Free Will, I omitted to provide an illustration that I think would be helpful.
My esteemed disputant has argued that "possibilities" in order to be "possible" must be possible in a libertarian sense. This is certainly not the case, but I failed to provide one of the easiest and best illustrations of this point in the heat of the debate.
The illustration is simple: in common speech we use "possibilities" to refer to things that we know full well are mechanically deterministic. Thus, for example, we speak about the possibility of drawing a "face card" as the next card in the deck, even though we know that it is already mechanically determined what card will be drawn next.
From our perspective, there are up to 52 possible next cards. In reality, only the actual card sitting on top of the deck will be drawn. The other 51 possibilities are not an illusion, they just reflect our ignorance.
The same kind of linguistic convention applies to our discussion about choice. Even if our choices are determined, we don't know what has been determined. Accordingly, from our perspective, there are alternative futures, although in reality God has already determined which of the two possibilities we will select.
This meshes well with my point in the debate that God takes as much credit for the outcome of "lots" (think dice, not real estate) and the choices of animals as God takes for human choices. In fact, God emphasizes his sovereignty in the last category even more than the other two areas.
Ultimately, as I established in the debate, the question is resolved by the fact that God states both that we choose and that God determines what we choose.