We know that our choices are predetermined because we, and everything in our world, was made by God, who knows all things and is also the author of time. He thus logically predetermines, and causes, our choices.
So then, how are they "real" choices?
That depends on what "real" means. If "real" means "not predetermined," then our predetermined choices are obviously not "real" choices. But if a "real choice" is when we see two different options in front of us and, because of our own desire or will, we select one of those options, then the fact that our will is created by God (and that our choices are predetermined) doesn't get in the way with this process at all.
Indeed, it makes it possible!
If our wills were never created, if our inclinations, preferences, and desires were not caused by God, then they would never exist at all. It is because God exists that we are able to willingly select the options we do.
But, if our choices are caused by God, how then are we accountable?
We are accountable because God says we are. He is the one who determines truth (John 14:6; Genesis 1:1; Isaiah 44:24-28). He says that if we disobey his prescriptive commands, we are guilty of sin (Romans 4:5; 1 John 3:4). And he says that ultimately, the penalty for sin is death (Romans 5:12, 6:23; Genesis 2:17).
Think about it this way. Who would we be accountable to if our choices are not part of God's creation? If our choices are not part of God's domain, how would he have any right to judge them? He would be an usurper!
Or, think of it another way. When Grand Moff Tarkin destroyed Alderaan in the Star Wars films, was he guilty? Did he do something sinful? Of course, we all agree that he did. But wait -- George Lucas caused it! George Lucas predetermined that Grand Moff Tarkin would make the choice he made! Okay, sure, but Grand Moff Tarkin still destroyed Alderaan. And he's still guilty as a result.
Generally, at this point, people tend to respond by yelling "Grand Moff Tarkin isn't real!"
But, again, that just depends on what "real" means.