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.
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