In the case of Riemannian manifolds, there are ways to take two manifolds and glue them together to get a new Riemannian manifold. For example, taking connected sums in local regions where the two metrics coincide. In the case of real hyperbolic surfaces, one can cut along geodesics and glue along these.
Complex manifolds are much more rigid objects, so I don't know if it is reasonable to expect gluing constructions to exist. Is anyone aware of such things?
I am most interested in the case of complex 2-manifolds. For example, is there a way to cut along real 3-manifolds inside a pair of complex 2-manifolds and glue to assemble a new complex 2-manifold?
I assume the integrability condition for upgrading almost-complex to complex must play a role somewhere, but I don't understand it well enough to have an intuition about how.