Thermodynamics often presumes that the observer has a macroscopically complete information about the system she or he deals with: no parasitic current, exact evaluation of the forces needed to drive the system out of equilibrium. However, most often the observer only measures marginal information. How is she or he to make consistent thermodynamic claims? Disregarding sources of dissipation might lead to bold claims, such as the possibility of perpetuum mobile. We show that it is nevertheless possible to produce an effective description that does not dispense with the fundamentals of thermodynamics: the 2nd Law, the Fluctuation-Dissipation paradigm, and the more recent and encompassing Fluctuation Theorem. We base our theory on the theory of the stochastic thermodynamics of nonequilibrium systems, introducing a class of "marginally time-reversed" Markovian generators that have interesting mathematical and physical properties.