  In this paper we study couples of finite separable extensions of the function
field $\mathbb{F}_q(T)$ which are arithmetically equivalent, i.e. such that
prime ideals of $\mathbb{F}_q[T]$ decompose with the same inertia degrees in
the two fields, up to finitely many exceptions. In the first part of this work,
we extend previous results by Cornelissen, Kontogeorgis and Van der Zalm to the
case of non-geometric extensions of $\mathbb{F}_q(T)$, which are fields such
that their field of constants may be bigger than $\mathbb{F}_q$. In the second
part, we explicitly produce examples of non-geometric extensions of
$\mathbb{F}_2(T)$ which are equivalent and non-isomorphic over
$\mathbb{F}_2(T)$ and non-equivalent over $\mathbb{F}_4(T)$, solving a
particular Inverse Galois Problem.
