  Josephson junctions (JJs), where both time-reversal and inversion symmetry
are broken, exhibit a phase shift $\phi_0$ in their current-phase relation.
This allows for an anomalous supercurrent to flow in the junction even in the
absence of a phase bias between the superconductors. We show that a finite
phase shift also manifests in the so-called Andreev interferometers - a device
that consists of a mesoscopic conductor coupled to the $\phi_0$-junction. Due
to the proximity effect, the resistance of this conductor is phase-sensitive -
it oscillates by varying the phase of the JJ. As a result, the quasiparticle
current $I_{\mathrm{qp}}$ flowing through the conductor has an anomalous
component, which exists only at finite $\phi_0$. Thus, the Andreev
interferometry could be used to probe the $\phi_0$ effect. We consider two
realizations of the $\phi_0$-junction and calculate $I_{\mathrm{qp}}$ in the
interferometer: a superconducting structure with spin-orbit coupling and a
system of spin-split superconductors with spin-polarized tunneling barriers.
