  For every integer $g \,\geq\, 2$ we show the existence of a compact Riemann
surface $\Sigma$ of genus $g$ such that the rank two trivial holomorphic vector
bundle ${\mathcal O}^{\oplus 2}_{\Sigma}$ admits holomorphic connections with
$\text{SL}(2,{\mathbb R})$ monodromy and maximal Euler class. Such a monodromy
representation is known to coincide with the Fuchsian uniformizing
representation for some Riemann surface of genus $g$. The construction carries
over to all very stable and compatible real holomorphic structures for the
topologically trivial rank two bundle over $\Sigma$ and gives the existence of
holomorphic connections with Fuchsian monodromy in these cases as well.
