Trinification from a complete $E_{6}$ GUT model

$\mathrm{E}_{6}$ Grand Unified Theories (GUTs) introduce novel symmetry-breaking patterns compared to the more common $\mathrm{SU}(5)$ and $\mathrm{SO}(10)$ GUT. We explore how $\mathrm{SU}(3)^3$ (trinification) or $\mathrm{SU}(6)\times\mathrm{SU}(2)$ symmetries can explicitly arise from $\mathrm{E}_{6}$ at an intermediate breaking stage. The representation $\mathbf{650}$ of $\mathrm{E}_{6}$ emerges as the lowest-dimensional candidate for breaking into one of the novel intermediate symmetries. Demanding subsequent breaking to the Standard Model group and a realistic Yukawa sector, we argue that the minimal ``realistic'' model of this type has the scalar sector $\mathbf{650}\oplus\mathbf{27}\oplus\mathbf{351'}$. Perturbativity curbs the construction of larger alternatives, so this model seems to be unique in its class. Assuming minimal tuning in scalar masses, three intermediate scenarios are consistent with unification: trinification $\mathrm{SU}(3)_C\times\mathrm{SU}(3)_L\times\mathrm{SU}(3)_R$ with either LR (left-right) or CR (color-right) parity, and $\mathrm{SU}(6)_{CR}\times\mathrm{SU}(2)_L$.