On the super-/supra- prefix (BTS, Nicolaou): I really believe this is part of the problem. Why should the extra-ordinary be construed as supra-ordinary, i.e. "above," "higher than" or "superior to" the ordinary (nature, or reason, etc.)? You might as well picture it below (as in depth) or besides (as in "fringe," "margin" or "limit" experience). Why hierarchise the difference?
Narkissos, you are going to hate me. More C.S. Lewis. The first I ever heard the term subnatural used was in "Miracles", Chapter 3 "The Cardinal Difficulty of Naturalism". I take it he is describing quantum phenomena, which act in a way that seems to betray reason. Merely posted here for your amusement.
http://www.philosophy.uncc.edu/mleldrid/Intro/csl3.html
One threat against strict Naturalism has recently been launched on which I myself will base no argument, but which it will be well to notice. The older scientists believed that the smallest particles of matter moved according to strict laws: in other words, that the movements of each particle were 'interlocked' with the total system of Nature. Some modern scientists seem to think--if I understand them--that this is not so. They seem to think that the individual unit of matter (it would be rash to call it any longer a 'particle') moves in an indeterminate or random fashion; [19] moves, in fact, 'on its own' or 'of its own accord'. The regularity which we observe in the movements of the smallest visible bodies is explained by the fact that each of these contains millions of units and that the law of averages therefore levels out the idiosyncrasies of the individual unit's behaviour. The movement of one unit is incalculable, just as the result of tossing a coin once is incalculable: the majority movement of a billion units can however be predicted, just as, if you tossed a coin a billion times, you could predict a nearly equal number of heads and tails. Now it will be noticed that if this theory is true we have really admitted something other than Nature. If the movements of the individual units are events 'on their own', events which do not interlock with all other events, then these movements are not part of Nature. It would be, indeed, too great a shock to our habits to describe them as super-natural. I think we should have to call them sub-natural. But all our confidence that Nature has no doors, and no reality outside herself for doors to open on, would have disappeared. There is apparently something outside her, the Subnatural; it is indeed from this Subnatural that all events and all 'bodies' are, as it were, fed into her. And clearly if she thus has a back door opening on the Subnatural, it is quite on the cards that she may also have a front door opening on the Supernatural-and events might be fed into her at that door too.