  The $g$-vectors of two-term presilting complexes are important invariants. We
study a fan consisting of all $g$-vector cones for a complete gentle algebra.
We show that any complete gentle algebra is $g$-tame, by definition, the
closure of a geometric realization of its fan is the entire ambient vector
space. Our main ingredients are their surface model and their asymptotic
behavior under Dehn twists. On the other hand, it is known that any complete
special biserial algebra is a factor algebra of a complete gentle algebra and
the $g$-tameness is preserved under taking factor algebras. As a consequence,
we get the $g$-tameness of complete special biserial algebras.
