Ah, thanks, maybe this holds a clue! (Clearly I have been interested in getting to the bottom of this for a while.)

So maybe an "algebraic effect" is one that's isomorphic to a free monad of a functor that itself is an algebraic data type. That seems to give an unambiguous specification for what it means to handle an effect (a natural transformation) and to take a "free product" of effects (sum the functors).

On the other hand I think it would mean that things like Future and general IO wouldn't be algebraic effects.