Reconstructing Structures with the Strong Small Index Property up to Bi-Definability

Let $\mathbf{K}$ be the class of countable structures $M$ with the strong small index property and locally finite algebraicity, and $\mathbf{K}_*$ the class of $M \in \mathbf{K}$ such that $acl_M(\{ a \}) = \{ a \}$ for every $a \in M$. For homogeneous $M \in \mathbf{K}$, we introduce what we call t