If the supernatural was to be the cause of something inside nature (ex: DNA, or miracle.) than it would be science who would determine it, not theology. So although science deals with nature, it does not have to be naturalistic.
Science uses something called (to make it sound fancy) "methodological Naturalism". This simply means that we use the approach of science to learn about things that can be observed*. If something tinkers with something "inside nature" it might be observable. Then science could comment on it.
There has never been
any way suggested of learning about anything that beats this approach. If we can't compare a suggestion to reality (observe something) then we have no other way of knowing if it is correct or not. NONE.
We know that there is an ocean of nonsense out there. If, submerged in that ocean there are tidbits of very interesting fact then it gets very hard to find them if we can't bring observation and testing to bear on it.
If you think there is any other way to reliably learn about anything then I suggest you start a thread and give us the details. Somehow no one comes forward with this.
The difficulty is, of course, that a god (as the word is generally used) can "tinker" in any number of ways that may not conform to any regularity or physical laws. If this doesn't happen to be captured at the moment of occurrence and never repeats there isn't much that can practically be done to learn about it. If then falls into the many, many "mmmm intersting"s that we have and can do nothing more with.
* an observation does not have to be (and in general should not be) only something we can see sitting in front of our noses. Since we don't actually "see" anything in a manner that most of us believe we do this makes sense. The word is misunderstood to a huge degree.