Seems some linguists use "naturalistic" -- but I'm not clear on what it means in this context or if it's different from "natural".
"Natural language" is often used in linguistics simply to mean language with some hint that it may be different from computer, animal or somehow artificial languages. It is most commonly found as part of the term "Natural Language Processing" which is basically a subfield of computational linguistics.
"Naturalistic language" is much less clearly defined and not really used in opposition to "natural language". It is often used to imply real, unrehearsed or unguided language - but still natural language (not artificial language made to look natural). The two common terms in which it is used are "naturalistic language learning" (meaning not in a classroom - and naturalistic refers to learning rather than language) and "naturalistic speech" (meaning live, unrehearsed speech).
"Natural language" refers to ordinary human languages that children learn: not programming languages or constructed languages. "Naturalistic speech" refers to speech that is "everyday", as opposed to high-falutin rhetorical style. You would have to investigate the context further, to determine if there is something in particular that is to be avoided as unnatural.