I have difficulties proving the following statement, which I think is used implicitly in Tent and Ziegler's book:
Let us work in a monster model, and assume that the formula $\varphi(x,b)$ divides over some set $A$ (i.e. there is some sequence $(b_i)_{i \in \omega}$ with $\text{tp}(b / A) = \text{tp}(b_i / A)$ for every $i$ and there is $k \in \omega$ such that the set $\left\{\varphi(x,b_i) \,|\, i \in \omega \right\}$ is $k$-inconsistent). Then there is some model $\mathfrak{M}$ with universe $M \supset A$, such that $\varphi(x,b)$ divides over $M$.
First, note that we can take the sequence $\mathcal{I} = (b_i)_{i\in\omega}$ witnessing that $\varphi(x,b)$ divides over $A$ to be $A$-indiscernible (this is Lemma 7.1.4 in Tent and Ziegler). Now let $M'\supset A$ be any model, and let $\mathcal{I}' = (b'_i)_{i\in\omega}$ be an $M'$-indiscernible sequence realizing $EM(\mathcal{I}/M')$ (by the "Standard Lemma" 7.1.1 in Tent and Ziegler). Since $\text{tp}(\mathcal{I}/A) = \text{tp}(\mathcal{I}'/A)$, there is an automorphism $\sigma$ fixing $A$ and moving $\mathcal{I}'$ to $\mathcal{I}$. Let $M = \sigma(M')$. Then $\mathcal{I}$ is $M$-indiscernible, so it witnesses that $\varphi(x,b)$ divides over $M$.