  We are interested in the stability of a class of totally geodesic wave maps,
as recently studied by Abbrescia and Chen, and later by Duan and Ma. The
relevant equations of motion are a system of coupled semilinear wave and
Klein-Gordon equations in $\mathbb{R}^{1+n}$ whose nonlinearities are critical
when $n=2$. In this paper we use a pure energy method to show global existence
when $n=2$. By carefully examining the structure of the nonlinear terms, we are
able to obtain uniform energy bounds at lower orders. This allows us to prove
pointwise decay estimates and also to reduce the required regularity.
