  According to Radzikowski's celebrated results, bisolutions of a wave operator
on a globally hyperbolic spacetime are of Hadamard form iff they are given by a
linear combination of distinguished parametrices
$\frac{i}{2}\big(\widetilde{G}_{aF} - \widetilde{G}_{F} + \widetilde{G}_A -
\widetilde{G}_R\big)$ in the sense of Duistermaat-H\"ormander. Inspired by the
construction of the corresponding advanced and retarded Green operator
$G_A,G_R$ as done in B\"ar, Ginoux, Pf\"affle 2007, we construct the remaining
two Green operators $G_F, G_{aF}$ locally in terms of Hadamard series.
Afterwards, we provide the global construction of
$\frac{i}{2}\big(\widetilde{G}_{aF} - \widetilde{G}_{F}\big)$, which relies on
new techniques like a well-posed Cauchy problem for bisolutions and a patching
argument using \v{C}ech cohomology. This leads to global bisolutions of
Hadamard form, each of which can be chosen to be a Hadamard two-point-function,
i.e. the smooth part can be adapted such that, additionally, the symmetry and
the positivity condition are exactly satisfied.
