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A three-terminal Josephson junction biased at opposite voltages can sustain aphase-sensitive dc-current carrying three-body static phase coherence, known asthe "quartet current". We calculate the zero-frequency current noisecross-correlations and answer the question of whether this current is noisy(like a normal current in response to a voltage drop) or noiseless (like anequilibrium supercurrent in response to a phase drop). A quantum dot with alevel at energy $\epsilon_0$ is connected to three superconductors $S_a$, $S_b$and $S_c$ with gap $\Delta$, biased at $V_a=V$, $V_b=-V$ and $V_c=0$, and withintermediate contact transparencies. At zero temperature, nonlocal quartets (inthe sense of four-fermion correlations) are noiseless at subgap voltage in thenonresonant dot regime $\epsilon_0/\Delta\gg 1$, which is demonstrated with asemi-analytical perturbative expansion of the cross-correlations. Noise revealsthe absence of granularity of the superflow splitting from $S_c$ towards$(S_a,S_b)$ in the nonresonant dot regime, in spite of finite voltage. In theresonant dot regime $\epsilon_0/\Delta< 1$, cross-correlations measured in the$(V_a,V_b)$ plane should reveal an "anomaly" in the vicinity of the quartetline $V_a+V_b=0$, related to an additional contribution to the noise,manifesting the phase sensitivity of cross-correlations under the appearance ofa three-body phase variable. Phase-dependent effective Fano factors $F_\varphi$are introduced, defined as the ratio between the amplitudes of phasemodulations of the noise and the currents. At low bias, the Fano factors$F_\varphi$ are of order unity in the resonant dot regime $\epsilon_0/\Delta<1$, and they are vanishingly small in the nonresonant dot regime$\epsilon_0/\Delta\gg 1$.

Gate-tunable zero-frequency current cross-correlations of the quartet mode in a voltage-biased three-terminal Josephson junction