  We predict a novel $(I_0,\phi_0)$-junction state of multi-terminal Andreev
interferometers that emerges from an interplay between long-range quantum
coherence and non-equilibrium effects. Under non-zero bias $V$ the
current-phase relation $I_S(\phi)$ resembles that of a $\phi_0$-junction
differing from the latter due to a non-zero average $I_0(V) = \left<
I_S(\phi)\right>_{\phi}$. The flux-dependent thermopower ${\mathcal S}(\Phi)$
of the system exhibits features similar to those of a $(I_0,\phi_0)$-junction
and in certain limits it can reduce to either odd or even function of $\Phi$ in
the agreement with a number of experimental observations.
