  We develop a detailed theory describing a non-trivial interplay between
non-equilibrium effects and long-range quantum coherence in superconducting
hybrid nanostructures exposed to a temperature gradient. We establish a direct
relation between thermoelectric and Josephson effects in such structures and
demonstrate that at temperatures exceeding the Thouless energy of our device
both phase-coherent thermoelectric signal and the supercurrent may be strongly
enhanced due to non-equilibrium low energy quasiparticles propagating across
the system without any significant phase relaxation. By applying a temperature
gradient one can drive the system into a well pronounced $\pi$-junction state,
thereby creating novel opportunities for applications of Andreev
interferometers.
