  Open classical systems with balanced, spatially separated gain and loss, also
called $\mathcal{PT}$ symmetric systems, are a subject of intense, ongoing
research. We investigate the properties of a classical chain with spatially
separated viscous loss and stochastic gain that are balanced only in a
statistical sense. For a purely harmonic chain, we show that a split Langevin
bath leads to either the absence of thermalization or non-equilibrium steady
states with inhomogeneous temperature profile. Both phenomena are understood in
terms of normal modes of the chain, where dissipation in one normal mode is
correlated with the velocities of all other modes. We obtain closed-form
expressions for the mode temperatures and show that nonlinearities lead to
steady states due to mode mixing even in the presence of a split Langevin bath.
