Since I found the answer to my question I will write it here, hoping that it will be helpful to someone else.
 
**Answer:** The ring of global real analytic functions on a compact real analytic manifold is noetherian due to Theorem I.9. [J.Frisch.: *Points de platitude d'un morphisme d'espaces analytiques complexes*. Invent.
Math., 4:118-138, 1967.], see also last paragraph in [J. M. Ruiz. *On Hilbert's 17th problem and real Nullstellensatz for global analytic functions*.
Math. Z., 190(3):447-454, 1985]