Let $\mathcal{M}$ and $\mathcal{N}$ be two $\mathcal{L}$-structures and suppose that for n-tupls $\bar{a}\in M^n$ and $\bar{b}\in N^n$, $tp^\mathcal{M}(\bar{a})=tp^\mathcal{N}(\bar{b})$ where $tp^\mathcal{M}(\bar{a})=\{\varphi: \mathcal{M}\models\varphi(\bar{a})\}$. Is it true that:
There is an $\mathcal{}L$-structure $\mathcal{A}$ and elementary embeddings $f:\mathcal{M}\hookrightarrow\mathcal{A}$ and $g:\mathcal{N}\hookrightarrow\mathcal{A}$ such that $f(a_i)=g(b_i)$ for $1\le i\le n$.
or there is a Counterexample?
Hint: Note that $(\mathcal M, \overline a)$ and $(\mathcal N, \overline b)$ are elementarily equivalent. This is just saying that $tp^{\mathcal M}(\overline a)=tp^{\mathcal N}(\overline b)$, in a different way. There is a standard proposition that then allows you to conclude.