There are some first-order theories $T$ with the property that for any $M \models T$, the only $M$-invariant global types are those that are coheirs over $M$ (i.e., finitely satisfiable in $M$). Two easy NIP examples of this are $\mathbb{Z}$ as a circular order and $2^{<\omega}$ as a partial order (by extension). This means that passing to coheirs is the only fully general method for constructing invariant types.
Nevertheless, other kinds of invariant types seem very common in unstable NSOP theories. Every NSOP theory I have thought carefully about seems to have many definable types. In unstable NSOP theories, these types are often not also coheirs (although they can be). I'm wondering if this is actually a general phenomenon or a symptom of lack of examples. There are plenty more specific questions one could ask, but I think it would be good to start with the asking for the easiest possible counterexample.
Question. Is there an unstable NSOP theory $T$ with the property that for any $M \models T$, every $M$-invariant global type is finitely satisfiable in $M$?
One important relevant fact is this: If $p(x)$ is an $M$-invariant type that is finitely satisfiable in some small model, then it is finitely satisfiable in $M$. I also have an argument that if there is a model $M\models T$ and an $M$-invariant type $p(x)$ that is not finitely satisfiable in $M$, then there is a model $N \preceq M$ with $|N| = |T|$ and an $N$-invariant type $q(x)$ that is not finitely satisfiable in $N$. (More generally, if $M \models T$, $\varphi(x,y)$ is a formula, and $b$ is some parameter in the monster such that there is an $M$-invariant type $p(x) \ni \varphi(x,b)$, then there is a club of $|T|$-sized elementary substructures of $M$ with the same property. Once you have this, you just need the observation that if $\varphi(x,b)$ is not satisfiable in $M$, it won't be satisfiable in any elementary substructure of $M$ either.)
I have been thinking about this for a while and I have made very little progress beyond this.
