Did you try to think of an example? Pick your favorite theory that's easy to understand, pick two models and a tuple from each satisfying the same type, and I bet you can amalgamate them.
Here's an example:
Let $V$ and $W$ be $\mathbb{Q}$-vector spaces, with bases $\\{v_1,\dots,v_n\\}$ and $\\{w_1,\dots,w_m\\}$. Then $(v_1,v_2)$ have the same type in $V$ as $(w_1,w_2)$ in $W$, and indeed we can find a vector space $S$ with basis $\\{s_1,\dots,s_{m+n-2}\\}$ and embeddings $V\to S$ and $W\to S$ defined by $v_1\mapsto s_1$, $v_2\mapsto s_2$, $v_i\mapsto s_i$ for $i<2$, and $w_1\mapsto s_1$, $w_2\mapsto s_2$, and $w_i \mapsto s_{n+i-2}$ for $i>2$. These maps identity $(v_1,v_2)$ with $(w_1,w_2)$ as desired, and they're elementary embeddings by quantifier elimination.