Definition: existing outside of nature; inexplicable by ordinary means
I can’t quite figure out how this is different from “supernatural.”
After some research, as near as I can tell, “preternaturally” means something extraordinary or explainable, but still within the bounds of physical possibility and reality. It’s often used to describe a high level of skill.
“Supernatural” is something clearly caused by or related to things beyond/above the natural world.
Why I Looked It Up
From a movie review blurb in the New York Times.
In “Last Night in Soho,” the lives of a modern-day fashion student and a singer from the 1960s preternaturally fuse, resulting in gorgeous horror.
That actually seems to be a mis-use of the term. Based on my understanding of the movie, it should be “supernaturally fuse.”