In addition to the relatively boring extension/restriction of scalars for $GSp(2n,\widetilde{k})$ for an extension field $\widetilde{k}$ of the ground field $k$... :
The Galois twist groups often denoted by $Sp^*(p,q)$ (or $GSp^*(p,q)$...) over ground field $\mathbb R$, defined via non-degenerate quaternion-symmetric forms with quaternion algebras over the ground field (or over extensions...) almost everywhere locally become $Sp(p+q,k_v)$, because the quaternion algebra splits almost everywhere locally. (Throwing in the similitudes is minor...)
That is, with quaternion (division) algebra $B$ over ground field $k$ (a global field), with non-degenerate $B$-valued, symmetric form on a $B$-vectorspace $V$ of $B$-dimension $n$, the isometry group $G$ has $k_v$-points isomorphic to $Sp(2n,k_v)$ locally almost everywhere.
EDIT: it may be worth adding that this sort of Galois twisting allows easy creation of compact quotients analogous to Shimura curves for $SL(2)=Sp(2)$, by choosing a symmetric form whose signature at one real place is $(0,q)$ or $(p,0)$, so that (by a standard, if not widely understood, reduction theory result) the arithmetic quotient is compact. But/and the representation theory is just that of (locally, split) $Sp(n)$ almost everywhere.