Free monad over an applicative?

Given an applicative f, what constructions are there that I can apply to f that make it a monad? Ideally, the monad would not be completely free, but satisfy a <*> b == ap a b. Does such a construction exist? If everything fails, I can always use Free f, but that will usually be “too free”. The reason I want that equation to hold is because my Applicative instance is very performant, and I don’t want to lose that by later slowly interpreting the free structure.

In my use case, I have a type operator t that has a very efficient singleton :: a -> t a and cartesian :: t a -> t b -> t (a, b), but no Functor. So I implemented:

data Coyoneda b = forall a . Coyoneda
  { t :: t a
  , f :: a -> b

Now I want to create a free monad on top of that. Maybe there is a shortcut that passes directly from t to a free monad that will reuse singleton and cartesian? The trouble with Free, operational etc. is that they will typically implement <*> = ap, which uses the free structure, but doesn’t reuse cartesian.

It seems that my question is answered here: Applicative Effects in Free Monads

There is a free monad over an applicative which uses the applicative structure: Control.Monad.Free.Ap