Science is not based on axioms, so contradictions require modifications to definitions and assumpptions. Self-consistency is certainly desirable, but science lives continuously with the unexplained and with apparent contradictions.
Then, based on what you said, science is often forced to cope with the supernatural: Logical self-consistency demands it.
Deeper still, science attempts to be consistent with the observed universe. If the universe is self-contradictory, so will science be. We tend to assume that the universe is internally consistent, but I highly doubt that such a thing is provable.