  Using field-theoretic techniques, we study the $SU(3)$ analogue of
anti-ferromagnetic Heisenberg spin model on the triangular lattice putting the
$p$-box symmetric representation on each site. Taking the large-$p$ limit, we
show that the low-energy effective theory is described by a $(2+1)$-dimensional
relativistic $SU(3)/U(1)^2$ nonlinear sigma model. Since the target space has a
nontrivial homotopy $\pi_2(SU(3)/U(1)^2)\simeq \mathbb{Z}^2$, this model has
two kinds of magnetic skyrmions, which can be created and annihilated by
monopole instantons. By careful analysis of the Wess-Zumino term in the spin
coherent path integral, we compute the Berry phase for these monopoles and it
produces the destructive interference. This restricts possible perturbations of
the effective Lagrangian by monopole operators, and we see that the
valence-bond-solid (VBS) phase should have degenerate ground states when
$p\not\in 3\mathbb{Z}$. We also compute 't Hooft anomalies to constrain
possible phases of this system, and a direct phase transition between N\'eel
and VBS phases is supported from the anomaly matching.
