Automorphic quotients for inner forms or $GSp(4)$

Let me give you the inner forms of $\mathrm{GSp}(4)$ with compact adelic quotient.

Every nonsplit inner form of $\mathrm{GSp}(4)$ is obtained in the following way: take $D$ a division quaternion algebra over $F$, let $V$ be a $D$-Hermitian space of $D$-dimension $2$, and construct $G = \mathrm{GU}(V)$.

By a theorem of Borel and Harish-Chandra, the adelic quotient of $G$ is compact if and only if $G$ modulo its center is anisotropic (in the algebraic group sense: it contains no nontrivial split torus).

This turns out to be equivalent to $V$ being anisotropic (in the quadratic/hermitian form sense: it has no nonzero isotropic vector). Over a number field, by the local-global principle for quadratic forms we can test anisotropy locally: over $p$-adic places, $V$ has $8$ variables as a quadratic form and is therefore isotropic, and over real places, $V$ is anisotropic if and only if it is positive definite or negative definite.

In summary, $G = \mathrm{GU}(V)$ has compact adelic quotient if and only if $F$ admits a real place at which $D$ is definite and $V$ is positive definite or negative definite.

For instance, the $D$-Hermitian form $x\bar{x}+y\bar{y}$ works over a totally definite quaternion algebra, but the one you wrote down is $x\bar{y}+y\bar{x}$.

As you write it, this group always has a non-trivial parabolic subgroup (the Siegel subgroup, given by the intersection of $GU_D$ with upper-triangular matrices in $GL_2(D)$); and hence its symmetric space cannot be compact.