$t$-Gauduchon Ricci-flat metrics on non-Kähler Calabi-Yau manifolds

We construct new examples of $t$-Gauduchon Ricci-flat metrics, for all $t<1$, on compact non-Kähler Calabi-Yau manifolds defined by certain principal torus bundles over rational homogeneous varieties with Picard number $\varrho (X)>1$. As an application, we provide a detailed description of new examples of Strominger-Bismut Ricci-flat Hermitian metrics, Lichnerowicz Ricci-flat Hermitian metrics, and balanced Hermitian metrics on principal $T^2$-bundles over the Fano threefold $\mathbb{P}(T_{{\mathbb{P}}^2})$.