This note comes in my lecture notes after the Löwenheim-Skolem Theorem and a remark on the equivalence of "$\Gamma \subseteq \text{Sent}(\mathcal{L})$ is strongly maximal consistent (i.e. for any $\psi\in\text{Sent}(\mathcal{L})$, then $\psi\in\Gamma$ or $\neg\psi\in\Gamma$)" and "$\Gamma = \text{Th}(\mathcal{A})$ for some $\mathcal{L}$-structure $\mathcal{A}$". Where $\mathcal{L}$ is a first order predicate language and $\text{Th}(\mathcal{A}):=\{\varphi \in \text{Sent}(\mathcal{L})|\mathcal{A}\models\varphi\}$.
I define $\Gamma$ to be maximal consistent if it is consistent and for any $\psi\in\text{Sent}(\mathcal{L})$, we have $\Gamma\vdash\psi$ or $\Gamma\vdash\neg\psi$.
Now, to come back to the question, I see the forward direction, using the fact that $\text{Th}(\mathcal{A})$ is maximal consistent, so if $\mathcal{A}$ and $\mathcal{B}$ are models for $\Gamma$, then $\Gamma\subseteq \text{Th}(\mathcal{A})\cap\text{Th}(\mathcal{B})$ implies equality.
For the backwards direction I already have the maximal consistency of $\text{Th}(\mathcal{A})$, which is a set of sentences independent from the choice of model $\mathcal{A}$ for $\Gamma$, but $\Gamma\subseteq\text{Th}(\mathcal{A})$, and here I stop.