  We theoretically study a simple non-equilibrium quantum network whose
dynamics can be expressed and exactly solved in terms of a time-local master
equation. Specifically, we consider a pair of coupled fermionic modes, each one
locally exchanging energy and particles with an independent, macroscopic
thermal reservoir. We show that the generator of the asymptotic master equation
is not additive, i.e. it cannot be expressed as a sum of contributions
describing the action of each reservoir alone. Instead, we identify an
additional interference term that generates coherences in the energy
eigenbasis, associated with the current of conserved particles flowing in the
steady state. Notably, non-additivity arises even for wide-band reservoirs
coupled arbitrarily weakly to the system. Our results shed light on the
non-trivial interplay between multiple thermal noise sources in modular open
quantum systems.
