I am interested in conditions under which there is a least model inside a (not saturated in general) model $N$ over certain configurations such as $M_0 \subseteq M_1, M_2$ with $M_1 \overset{\vert}{\smile}_{M_0} M_2$. The situation is simpler in an uncountably categorical theory, since $N$ has to be saturated and the intersection of all models containing $M_1M_2$ is $acl(M_1M_2)$.

*Question*: given $M_0 \subseteq M_1, M_2$ models with $M_1 \overset{\vert}{\smile}_{M_0} M_2$ inside the monster model of an uncountably categorical theory, is $acl(M_1M_2)$ a model?

On the face of it I don't see many reasons why it should be true. However, it is certainly true in almost strongly minimal theories and also in the classical example of an uncountably categorical theory that is not almost strongly minimal (the theory of $\bigoplus_{i < \omega} \mathbb Z/4\mathbb Z$).