I would like to characterize the theories that satisfy the following property for every formulas $\varphi(x)$ and $\psi(x)$ with parameters in the monster model and for every set of parameters $A$
If $\varphi(x)\vee\psi(x)$ is almost satisfied over $A$ then either $\varphi(x)$ or $\psi(x)$ is almost satisfied over $A$.
Almost satisfied over $A$ means: satisfied (i.e. consistent) in every model containing $A$. (To avoid trivialities, note that the model need not contain the parameters of the formulas.)
EDIT A similar property appears in an old paper of Harnik and Harrington, see Lemma 4.8 (the notation is heavy, I may be mistaken.) They call it fundamental lemma. I have never seen this lemma elsewhere. Its role could have been replaced by stronger facts. I would also love to see a different proof of it (maybe a proof using forking?). However, the property so natural that ought to be of interest in itself.