  We discuss the response of both moving and trapped solitary wave solutions of
a nonlinear two-component nonlinear Schr\"odinger system in 1+1 dimensions to
an anti-$\mathcal{PT}$ external periodic complex potential. The dynamical
behavior of perturbed solitary waves is explored by conducting numerical
simulations of the nonlinear system and using a collective coordinate
variational approximation. We present case examples corresponding to choices of
the parameters and initial conditions involved therein. The results of the
collective coordinate approximation are compared against numerical simulations
where we observe qualitatively good agreement between the two. Unlike the case
for a single-component solitary wave in a complex periodic
$\mathcal{PT}$-symmetric potential, the collective coordinate equations do not
have a small oscillation regime, and initially the height of the two components
changes in opposite directions often causing instability. We find that the
dynamic stability criteria we have used in the one-component case is proven to
be a good indicator for the onset of dynamic instabilities in the present
setup.
