  We consider Reaction-Diffusion systems on $\mathbb{R}$, and prove diffusive
mixing of asymptotic states $u_0(kx - \phi_{\pm}, k)$, where $u_0$ is a
periodic wave. Our analysis is the first to treat arbitrarily large
phase-offsets $\phi_d = \phi_{+}- \phi_{-}$, so long as this offset proceeds in
a sufficiently regular manner. The offset $\phi_d$ completely determines the
size of the asymptotic profiles, placing our analysis in the large data
setting. In addition, the present result is a global stability result, in the
sense that the class of initial data considered are not near the asymptotic
profile in any sense. We prove global existence, decay, and asymptotic
self-similarity of the associated wavenumber equation. We develop a functional
framework to handle the linearized operator around large Burgers profiles via
the exact integrability of the underlying Burgers flow. This framework enables
us to prove a crucial, new mean-zero coercivity estimate, which we then combine
with a nonlinear energy method.
