  The classical Cahn-Hilliard (CH) equation corresponds to a gradient dynamics
model that describes phase decomposition in a binary mixture. In the spinodal
region, an initially homogeneous state spontaneously decomposes via a
large-scale instability into drop, hole or labyrinthine concentration patterns
of a typical structure length followed by a continuously ongoing coarsening
process. Here we consider the coupled CH dynamics of two concentration fields
and show that nonreciprocal (or active, or nonvariational) coupling may induce
a small-scale (Turing) instability. At the corresponding primary bifurcation a
branch of periodically patterned steady states emerges. Furthermore, there
exist localized states that consist of patterned patches coexisting with a
homogeneous background. The branches of steady parity-symmetric and
parity-asymmetric localized states form a slanted homoclinic snaking structure
typical for systems with a conservation law. In contrast to snaking structures
in systems with gradient dynamics, here, Hopf instabilities occur at
sufficiently large activity which result in oscillating and traveling localized
patterns.
