Let $M$ be a real analytic variety
(if someone is concerned about distinction between
"real analytic spaces" and "real analytic varieties"
in real analytic geometry, let's assume that $M$ 
is both "variety" and "space"). I was sure it is
well-known that higher cohomology of any 
real analytic coherent sheaf over $M$ vanish.


The argument is standard: by Grauert, a real analytic variety is the set
of real points in a Stein variety $M\_{C}$ with an anticomplex involution $v$. A coherent sheaf over $M$ is the set of $v$-invariant sections 
of a $v$-equivariant coherent sheaf $F\_{C}$ on $M\_{C}$, and $F\_C$ has
vanishing cohomology because $M\_C$ is Stein.


Recently I needed a reference to this fact, and
I could not find it. I would be extremely grateful
for a reference! 



Cohomology of real analytic coherent sheaves
In a smooth case, the reference is Proposition 2.3 in Atiyah and Hirzebruch's [Analytic cycles on complex manifolds](https://www.sciencedirect.com/science/article/pii/0040938362900940).


For a non-smooth case, I don't know the general reference, but Theoreme 3 in Henri Cartan's paper [Variétés analytiques réelles et variétés analytiques complexes](http://www.numdam.org/article/BSMF_1957__85__77_0.pdf)
establishes theorems A and B for *coherent* (supports of coherent analytic sheaves) analytic subvarieties of $\mathbb{R}^n$


