  We present an avatar of the Euler obstruction to foliated structures on
certain non-metric surfaces. This adumbrates (at least for the simplest
2D-configurations) that the standard mechanism---to the effect that the devil
of algebra sometimes barricades the existence of angelic geometric structures
(obstruction theory more-or-less)---propagates slightly beyond the usual
metrical proviso. Alas, the game is much more conservative than revolutionary:
in particular we enjoyed retrospecting at Poincar\'e's argument of 1885
(announced in 1881).
