  Let W -> X be a real smooth projective 3-fold fibred by rational curves. J.
Koll\'ar proved that, if W(R) is orientable, then a connected component N of
W(R) is essentially either a Seifert fibred manifold or a connected sum of lens
spaces. Our Main Theorem, answering in the affirmative three questions of
Koll\'ar, gives sharp estimates on the number and the multiplicities of the
Seifert fibres and on the number and the torsions of the lens spaces when X is
a geometrically rational surface. When N is Seifert fibred over a base orbifold
F, our result generalizes Comessatti's theorem on smooth real rational
surfaces: F cannot be simultaneously orientable and of hyperbolic type. We show
as a surprise that, unlike in Comessatti's theorem, there are examples where F
is non orientable, of hyperbolic type, and X is minimal. The technique we use
is to construct Seifert fibrations as projectivized tangent bundles of Du Val
surfaces.

Algebraic Geometry;Geometric Topology