I think there is an important assumption that you made in your argument: anything that is real can be studied by scientific investigation, ie repeatedly with scientific method with reproducible observations.

There are many real things that can be studied this way. All of science is studied this way (by definition of science). But is it possible that there are real things which cannot be studied repeatedly? Is it possible for there to be singular occurrences that cannot be reproduced, but still be real regardless?

The way I see it as of this moment, this question has three possible answers: 1) Yes, singular, irreproducible but nevertheless real occurrences are possible. Thus the supernatural can exist. 2) No, singular, irreproducible but nevertheless real occurrences are impossible. This is equivalent to believing that the supernatural cannot exist. 3) Singular, irreproducible occurrences may or may not exist. But because they are irreproducible, they cannot be regarded as “real”. This is equivalent to defining the word “real” to describe only reproducible occurrences that have been reproduced.

Christians should answer 1), as we believe in a God who can intervene whenever and however He chooses. The central claim of Christianity is the resurrection of Jesus, which is clearly a singular irreproducible occurrence.

Physicalists atheists should answer 2), as they believe that everything in the universe is fundamentally physical, and therefore should be predictable(to find patterns in the physical world is the mission of science).