Logical systems that are complete but not sound

If a logical system is not sound, this means that there is a mis-match between the deductive system and the semantics (the models used when defining soundness). A typical example would be that, if we use Kripke structures as our semantics (so the appropriate logic would be intuitionistic), then classical logic is unsound. As far as I can see, such a phenomenon is not "useful" but just a mistake. (Well, I suppose it could be useful for showing that someone made a mistake.)