For a quaternion algebra $D$, introduce the quaternionic similitude unitary groups: \begin{equation} \mathrm{GU}_D = \left\{ g \in \mathrm{GL}(D) \ : \ g^\star \left( \begin{array}{cc} & 1 \\ 1 & \end{array} \right) g = \mu(g) \left( \begin{array}{cc} & 1 \\ 1 & \end{array} \right), \mu(g) \in \mathbf{G}_m \right\} \end{equation}

The $\mathrm{GU}_D $ are the inner forms of $\mathrm{GSp(4)}$ when $D$ describes the quaternion algebras. For almost every places $v$ of $F$, more precisely the split ones, this group is $G_v \simeq \mathrm{GSp(4,F_v)}$.

When are those $\mathrm{GU}_D$ of compact automorphic quotient?

I believe assuming $D$ to be totally definite at infinity (i.e.ramified at archimedean places) is enough, but is it necessary?