a disjunction property 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 Its role could have been replaced by stronger facts.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.
 A: By Lemma 3.12 in Baldwin's Fundamentals of Stability Theory, in a stable theory, a formula is almost satisfied over $A$ if and only if it does not fork over $A$. In fact, this is taken as the definition of non-forking in Harnik's paper Generic formulas and types a la Hodges. A disjunction of forking formulas forks (in any theory). So, as Primo Petri claimed in the comments, your disjunction property holds in all stable theories. 
Your property also holds in any theory with definable Skolem functions, since then a formula is almost satisfied over $A$ if and only if it is satisfied by some element of $\mathrm{dcl}(A)$. So for example the property holds in $\text{RCF}$. The same argument shows that your property holds in any theory in which $\mathrm{acl}(A)$ is a model for any set $A$. 
These two reasons for the property holding (stability and definable Skolem functions) have a different enough character that I suspect it would be difficult to give a satisfying characterization of theories with your property.  
