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The structure formation in peritectic Al-4.5at.%Cu-11at.%Ni ternary alloy with four-phase peritectic reaction was investigated using the quantitative phase-field model of eutectic growth. This model, extended to an arbitrary number of phases, guarantees the stability requirements on individual interfaces. The thermal noise terms disturb the stability and produce the heterogeneous nucleation of secondary phases in accordance to the energetic and concentration conditions. In our recent work it was shown that in differential thermal analysis (DTA) experiments specific microstructure parts in Al-4.5at.%Cu-11at.%Ni alloy with a four-phase peritectic reaction were observed, which cannot be explained by Scheil calculation or simple phase-field modeling. In this work, it was found by numerical experiments that, due to the formation of anisotropic quasi-primary Al3Ni2 crystals and the suppression of the nucleation of (Al) phase, the eutectic-like coupled growth of Al3Ni2 and Al3Ni phases can be observed. In addition, at further cooling the anisotropic shape of quasi-secondary Al3Ni crystals promotes the nucleation of the (Al) phase. The simulated final structure is comparable to the experimental one which is characterized by large Al3Ni2 crystals enveloped by the Al3Ni phase.