Automorphisms between two tuples of the same type

Question

Fix some first order structure in some fixed language. If tuple $b$ is the image of tuple $a$ under some automorphism, then they have the same type. Is the converse true? That is, for any fixed tuples $a$ and $b$ of the same type, is there an automorphism mapping $a\mapsto b$?

If so, do $a$ and $b$ need to satisfy the same first order formulas, or does it suffice to satisfy the same atomic formulas, or something like this?

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Edit: The question as stated above is false (see below). I am now wondering whether the substructure generated by $a$ and the substructure generated by $b$ are isomorphic

Answer 1

From Model theory: An introduction by David Marker, exercice 4.5.1:

Let $\mathcal{M}=(X,<)$ be a dense linear order and $A= \{1- \frac{1}{n} \mid n \geq 1 \} \cup \{2+ \frac{1}{n} \mid n \geq 1 \}$. Show that $1$ and $2$ realize the same type over $A$, but there is no automorphism of $\mathbb{Q}$ fixing $A$ pointwise sending $1$ to $2$.

And exercice 4.5.9:

Let $x$ and $y$ be algebraically independent over $\mathbb{R}$. Order $\mathbb{R}(x,y)$ such that $x>r$ for all $r \in \mathbb{R}$ and $y>x^n$ for all $n>0$. Let $F$ be the real closure of $\mathbb{R}(x,y)$. Show that $\text{tp}^F(x)=\text{tp}^F(y)$, but there is no automorphism of $F$ sending $x$ to $y$.