  Consider a monad on an idempotent complete triangulated category with the
property that its Eilenberg-Moore category of modules inherits a triangulation.
We show that any other triangulated adjunction realizing this monad is
'essentially monadic', i.e. becomes monadic after performing the two evident
necessary operations of taking the Verdier quotient by the kernel of the right
adjoint and idempotent completion. In this sense, the monad itself is
'intrinsically monadic'. It follows that for any highly structured ring
spectrum, its category of homotopy (a.k.a. naive) modules is triangulated if
and only if it is equivalent to its category of highly structured (a.k.a.
strict) modules.
