  In this article, we study the obstructions to the local-global principle for
homogeneous spaces with connected or abelian stabilizers over finite extensions
of the field $\mathbb{C}((x,y))$ of Laurent series in two variables over the
complex numbers and over function fields of curves over $\mathbb{C}((t))$. We
give examples that prove that the usual Brauer-Manin obstruction is not enough
to explain the failure of the local-global principle, and we then construct a
variant of this obstruction using torsors under quasi-trivial tori which turns
out to work. In the end of the article, we compare this new obstruction to the
descent obstruction with respect to torsors under tori. For that purpose, we
use a result on towers of torsors, that is of independent interest and
therefore is proved in a separate appendix.
