A detailed numerical and experimental study of the ordering of the low-temperat ure tetragonal phase of CuAu is presented. The numerical simulations are based on a coarse-grained free energy derived from electronic structure calculations of CuAu, while the experimental results are obtained from in situ X-ray scattering. Both theoretical and experimental work indicate a subtle kinetic competition between the ordered tetragonal phase and the metastable modulated phase.