Yes, true and false exist perfectly in the real world. For example, an ant is never not an ant at the same time and in the same way as when its an ant.
Philosophically speaking, this falls in the category of British Empiricism.
Ergo, our senses give us our only knowledge of the external universe.
Be are born tabula rasa.
Our environment forms us.
Our environment consists of our parents, our friends, our teachers, our ministers, our schools, and our employers.
Something either exists or it does not exist.
And then there is the vast unknown.
For example, the half dozen or more classic proofs of God suggest there is a God. But we do not know it nor can we prove it.
We know that we exist. Rene Descartes taught us this as well as Cartesian Geometry.
It is a safe bet that others exist also, since they can give us joy and pain, and even kill us.
Any statement is either valid or invalid.
Therefore truth exists objectively.
Something is either true or false.
There are no unicorns. At least none that we have ever seen in 10,000 years of human civilization.