  We consider a non-reciprocically coupled two-field Cahn-Hilliard system that
has been shown to allow for oscillatory behaviour, a suppression of coarsening
as well as the existence of localised states. Here, after introducing the model
we first briefly review the linear stability of homogeneous states and show
that all instability thresholds are identical to the ones for a corresponding
Turing system (i.e., a two-species reaction-diffusion system). Next, we discuss
possible interactions of linear modes and analyse the specific case of a
``Hopf-Turing'' resonance by discussing corresponding amplitude equations in a
weakly nonlinear approach. The thereby obtained states are finally compared
with fully nonlinear simulations for a specific conserved amended
FitzHugh-Nagumo system. We conclude by a discussion of the limitations of the
weakly nonlinear approach. The published version of this preprint can be found
under T. Frohoff-Hu\"ulsmann, U. Thiele and L. M. Pismen. Nonreciprocity
induces resonances in two-field Cahn-Hilliard model. Phil. Trans. R. Soc. A.
381: 20220087. 20220087. DOI: 10.1098/rsta.2022.0087
