In order to test the monadicity of a functor, there is a *precise monadicity theorem* (PM) as well as a *crude monadicity theorem* (CM), see the nlab. In CM, the forgetful functor should create reflexive coequalizers.

What is a nice and specific example which shows that CM is only sufficient, but not necessary; i.e. what is a monad which doesn't preserve reflexive coequalizers? I couldn't find such an example.

Besides, is the condition in CM satisfied in almost all practical / non-pathological examples? For example it holds for algebraic monads on $\mathsf{Set}$. What is the intuitive difference between PM and CM? The background for this rather soft question is the following: I would like to prove a certain theorem about certain monads, and in one step it would be very convenient if the monad preserved reflexive coequalizers. Now I wonder if there is any problem when I just add this assumption to the theorem.

Given a monad defined by generators and relations (on a category different from $\mathsf{Set}$), how can I test practically if the forgetful functor creates reflexive coequalizers?