  We develop a theory of canonical isogeny characters of Drinfeld-Stuhler
modules similar to the theory of canonical isogeny characters of abelian
surfaces with quaternionic multiplication. We then apply this theory to give
explicit criteria for the non-existence of rational points on Drinfeld-Stuhler
modular varieties over the finite extensions of $\mathbb{F}_q(T)$. This allows
us to produce explicit examples of Drinfeld-Stuhler curves violating the Hasse
principle.
