"IF something were truly supernatural, it wouldn't necessarily register on any scientific equipment..."
but then we would also be unable to percieve it at all. even if that "supernatural something" is utterly undetectable of and by itself, it must have some physical, or at least empirical, effects. and those CAN be investigated
one study of NDEs/OBEs -- utterly subjective experiences -- looked at what the subject was able to observe. in the case in mind, there was a randomly-generated word on a laptop that could *only* be seen from an near-the-ceiling perspective. if the subject did indeed "float above his/her body", then the randomly-generated word would have been visible, and the subject could identify it. hasn't happened yet, but when/if it does, that would be objective evidence of a subjective event