  We study the canonical complexifications of non-compact Riemannian symmetric
spaces G/K by the Grauert tube construction. We determine the maximal such
complexification, a domain already constructed in another context by Akhiezer
and Gindikin (Math. Ann., 1990), and show that this domain is Stein. We show
there is an alternative for a G-invariant complexification: it is either
"rigid" (its automorphism group is G), or it is a Hermmitian symmetric space.
We also determine when invariant complexifications, especially the maximal one,
are Hermitian symmetric. This is expressed simply in terms of the ranks of the
symmetric spaces involved.
