What kind of additional properties and/or structures one needs to impose on the category of (commutative or noncommutative) monoids of some monoidal category so that one can recover the original monoidal category from this data?

What kind of additional properties and/or structures one needs to impose on a category to ensure that it is the category of monoids of some monoidal category?

The example I have in mind is the category of (commutative or noncommutative) C*-algebras (or von Neumann algebras). Can we obtain one of these categories as the category of monoids of some monoidal category?