  It is shown that if $X$ is an Inoue surface of type $S_M$ then the
irreducible components of the Douady space of $X^n$ are compact, for all $n>0$.
This gives an example of an essentially saturated compact complex manifold (in
the sense of model theory) that is not of Kaehler-type. Among the known compact
complex surfaces without curves, it is shown that these are the only examples.
