  We consider the higher-order resummation of Sudakov double logarithms in the
presence of multiple coupled gauge interactions. The associated evolution
equations depend on the coupled $\beta$ functions of two (or more) coupling
constants $\alpha_a$ and $\alpha_b$, as well as anomalous dimensions that have
joint perturbative series in $\alpha_a$ and $\alpha_b$. We discuss possible
strategies for solving the system of evolution equations that arises. As an
example, we obtain the complete three-loop (NNLL) QCD$\otimes$QED Sudakov
evolution factor. Our results also readily apply to the joint higher-order
resummation of electroweak and QCD Sudakov logarithms.
  As part of our analysis we also revisit the case of a single gauge
interaction (pure QCD), and study the numerical differences and reliability of
various methods for evaluating the Sudakov evolution factor at higher orders.
We find that the approximations involved in deriving commonly used analytic
expressions for the evolution kernel can induce noticeable numerical
differences of several percent or more at low scales, exceeding the
perturbative precision at N$^3$LL and in some cases even NNLL. Therefore, one
should be cautious when using approximate analytic evolution kernels for
high-precision analyses.
