What is the true wild Dianthus caryophyllus is a very compicated topic. I can't find any recent scientific paper on this subject.
Dianthus caryophyllus has been cultivated for such a long time and there is so much confusion with all the species, subspecies, varieties, synonyms and so on of all the populations of this group. A month ago, I wars reading the papers of Domina about Dianthus virgineus. At first glance, it clarify a lot of things, at the second lecture, I've even more confused than before.
Back to caryophyllus : as a carnation was already cultivated at greek ages, you can read that it originates from Greece or Middle East. I have no opinion about that. In France (
https://siflore.fcbn.fr/?cd_ref=94717&r=metro), some wild forms are reported to grow on some castles, old walls... I've never seen it and I'm also looking for a wild form of this species. If someone grow it, I would also appreciate a very few seeds.