Existence of nearly holomorphic sections on compact Hermitian symmetric spaces

Let $X=U\mathrm{/}K$ be a compact Hermitian symmetric space, and let $\text{sE}$ be a $U$-homogeneous Hermitian vector bundle on $X$. In a previous paper, we showed that the space of nearly holomorphic sections is well-adapted for harmonic analysis in $L^2(X,\text{sE})$ provided that non-trivial nearly holomorphic sections do exist. Here we investigate the problem of extending local nearly holomorphic sections to global ones and prove the existence of non-trivial nearly holomorphic sections. This extends the results on the $U$-type decomposition of $L^2(X,\text{sE})$ from our previous paper.