$\exists^\infty$-elimination and model companion. In the book Ziegler, Tent: A course in model theory it states

Exercise 5.5.7. Let $T_1$ and $T_2$ be two model complete theories in disjoint
  languages $L_1$ and $L_2$. Assume that both theories eliminate $\exists^\infty$ . Then $T_1\cup T_2$ has a model companion.

I want to prove this statement. Sine $T_1\cup T_2$ also eliminates $\exists^\infty$ (i.e. in all models realisation sets have either a finite upper bound or are infinite) my strategy was to use the previous exercise

Exercise 5.5.6. Assume that T eliminates the quantifier $\exists^\infty$. Then for every formula $\varphi(x_1 , . . . , x_n , \overline{y})$ there is a formula $\theta(\overline{y})$ such that in all models $\mathfrak{M}$ of
  $T$ a tuple $\overline{b}$ satisfies $\theta(\overline{y})$ if and only if $\mathfrak{M}$ has an elementary extension $\mathfrak{M}'$ with
  elements $a_1 , . . . , a_n\in M'\setminus M$ such that $\mathfrak{M}\vDash\varphi(a_1,...,a_n)$.

I define the theory 
$(T_1\cup T_2)^*:=\{\theta\;|\;$there exists a $L_1\cup L_2$-formula $\varphi(x_1,..x_n)$, that is satisfiable in all models $\mathfrak{M}\vDash T_1\cup T_2$ such that $\mathfrak{M}\vDash\theta\Leftrightarrow$ there exists an elementary extension $\mathfrak{M}'$ with
elements $a_1 , . . . , a_n\in M'\setminus M$ such that $\mathfrak{M}\vDash\varphi(a_1,...,a_n)$$\}$
To me it seemed to be right choice for the model companion of $T_1\cup T_2$ until I tried to prove that $(T_1\cup T_2)^*$ is model complete (without success). Does this choice make sense?
 A: Since $T_1$ and $T_2$ are model complete, they are $\forall\exists$-axiomatizable, and $T_1\cup T_2$ is also $\forall\exists$-axiomatizable. So to show the model companion exists, you need to axiomatize the existentially closed models of $T_1\cup T_2$. 
So suppose $M\subseteq M'$, both models of $T_1\cup T_2$. We want to write down a sufficient condition for $M$ to be existentially closed in $M'$. Let $\psi(x)$ be a quantifier-free formula with parameters from $M$ such that $M'\models \exists x\, \psi(x)$. You want to observe the following things:


*

*It suffices to assume that $\psi(x)$ is a conjunction of atomic and negated atomic $(L_1\cup L_2)$-formulas. 

*So if $L_1$ and $L_2$ are relational, $\psi(x)$ is actually a conjunction $\varphi_1(x)\land \varphi_2(x)$, where $\varphi_1$ is an $L_1$-formula and $\varphi_2$ is an $L_2$-formula. If the languages aren't relational, there's an issue that $\psi$ could mention terms formed using function symbols from both $L_1$ and $L_2$. Then you have to think about replacing $\psi$ with another formula obtained by "unnesting" all terms. It might be better to work out the details in the relational case first and then come back to this complication. 

*It suffices to assume that there is a witness $M'\models \psi(a')$ such that each element of $a'$ is in $M'\setminus M$ and the elements of $a'$ are all distinct. 
This leads us to the following axiomatization: Let $\varphi_1(x,y)$ be a quantifier-free $L_1$-formula, and let $\varphi_2(x,z)$ be a quantifier-free $L_2$-formula (here $x$, $y$, $z$ are tuples of variables). Let $\theta_1(y)$ and $\theta_2(z)$ be the formulas provided by Exercise 5.5.6. for $\varphi_1$ and $\varphi_2$, respectively. Let $\theta'_1(y)$ be the conjunction of $\theta_1(y)$ and inequations $y_i\neq y_j$ for $i\neq j$, and similarly for $\theta'_2(z)$. Then look at the following sentence: $$\forall y\, \forall z\, ((\theta'_1(y)\land \theta'_2(z))\rightarrow \exists x\, (\varphi_1(x,y)\land \varphi_2(x,z))).$$
The model companion of $T_1\cup T_2$ is axiomatized by $T_1\cup T_2$ together with all sentences of the above form. The discussion above can be viewed as an extended hint that every model of this theory is an existentially closed model of $T_1\cup T_2$. You also need to show the converse: that every existentially closed model of $T_1\cup T_2$ satisfies these extra axioms. Explicitly, given $\varphi_1(x,y)$ and $\varphi_2(x,z)$, if $M\models \theta_1'(b)\land \theta_2'(c)$, then you can embed $M$ in a model $M'$ of $T_1\cup T_2$ such that $M'\models \varphi_1(a',b)\land \varphi_2(a',c)$ for some $a'\in M'$, and use the fact that $M$ is existentially closed to find a witness in $M$. 
Aside: The statement of the exercise is a result from Peter Winkler's 1975 PhD thesis. It's a nice coincidence that you posted this problem now. Minh Tran, Erik Walsberg, and I are working on a project that we call "Interpolative Fusions" which at its heart is a generalization of this result of Winkler's. We just posted the first paper from this project to the arXiv this week: https://arxiv.org/abs/1811.06108
