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The word ''quasi-judicial'' conjures up an image of a genetically modified hybrid, comprising of half of a real Judge and half of something else - such as an extra-terrestial entity.
The word should only be used by Professor Wade or Craig (the Dons of Administrative law) not to describe a mediators role which is as you (correctly) noted far from judicial.
A supervisory body can easily be set up for mediation (if there is enough will) - the Meditators Regulation Board or the Mediators Regulation Authority.