Recently the question of whether religion is automatically supernatural in nature and whether it is possible to deny the supernatural and still be Christian came up on Roger Olson's Patheos blog. It was an odd piece in that I found myself somewhat impressed by Olson's immediate admission that Christianity and religion are, or at least contain elements of, supernatural.
I was rather less impressed with the reason it came up and how he set about writing about the connection. He seems to have just assumed that one of his readers assertion about a notable theologians position on the subject was precisely what the reader claimed it to be. This reader claimed that Karl Barth did not believe in the "supernatural" and insisted that Christianity did not require it. I am by no means an expert on Barth's work but I have read some of it and don't recall his making such a definitive claim. Not only does Olson not indicate that he bothered to review any of Barth's writing to determine if the claim had any merit (being dead we can't just ask him), he spends most of the piece trying to reconcile Barth's supposed view with his own.
Though the mental gymnastics were amusing it seemed rather futile. Even if Barth did make such a claim it would not be the first time a theologian had deluded himself into believing something that is blatantly incoherent and contradictory. There is no aspect of Christ's supposed life and ministry that is not laced with the supernatural. Miracles, for instance, are by definition outside the natural laws. So is one of the most important aspects of the Christ narratives: resurrection.
Basically, whether Barth believed that religion, specifically Christianity, was innately supernatural does not change the fact that Christianity has to be. Without the miracles, the resurrection, and the transfer of sins as a result Christ would be just another itinerant preacher/reformer.