  Let $X=U/K$ be a compact Hermitian symmetric space, and let $\sE$ 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 $L^2(X,\sE)$ 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
$L^2(X,\sE)$ from our previous paper.
