  Let $(N,c_N)$ be a complex vector bundle equipped with a real structure over
a real curve $(\Sigma,c_\Sigma)$ of genus $g \geq 0$. We compute the sign of
the action of the automorphisms of $(N,c_N)$ lifting the identity of $\Sigma$
on the orientations of the determinant line bundle over the space of real
Cauchy-Riemann operators on $(N,c_N)$. This sign can be obtained as the product
of two terms. The first one computes the signature of the permutations induced
by the automorphisms acting on the $Pin^\pm$ structures of the real part of
$(N,c_N)$. The second one comes from the action of the automorphisms of
$(N,c_N)$ on the bordism classes of real Spin structures on
$(\Sigma,c_\Sigma)$.
