© 2019 Author(s). We used transmission electron microscopy, Raman, and photoluminescence spectroscopy to identify the effect of CuPt-type GaP-InP atomic ordering (AO) on the structural and emission properties of self-organized (SO) InP/GaInP2 Wigner molecule (WM) quantum dot (QD) structures. We found that the correlation of AO and SO growth results in the formation of InP/GaInP2 QD/AO-domain (QD/AOD) core-shell composites. This observation shows that intrinsic WMs in this system emerge due to a strong piezoelectric field generated by AODs, which induces QD doping and a built-in magnetic field. We found that the bond relaxation of AODs leads to a decrease in the emission energy of WMs of 80 meV. The photoluminescence spectra of single WMs having an emission energy ~1.53 eV are presented here, the lowest one reported for this system.