  In this note we investigate the structure of the space $\Jj$ of smooth almost
complex structures on $S^2\times S^2$ that are compatible with some symplectic
form. This space has a natural stratification that changes as the cohomology
class of the form changes and whose properties are very closely connected to
the topology of the group of symplectomorphisms of $S^2\times S^2$.
  By globalizing standard gluing constructions in the theory of stable maps, we
show that the strata of $\Jj$ are Fr\'echet manifolds of finite codimension,
and that the normal link of each stratum is a finite dimensional stratified
space. The topology of these links turns out to be surprisingly intricate, and
we work out certain cases. Our arguments apply also to other ruled surfaces,
though they give complete information only for bundles over $S^2$ and $T^2$.
