Understanding the spatial organisation of the genome in the cell nucleus is one of the current grand challenges in biophysics. Certain biochemical -or epigenetic -marks that are deposited along the genome are thought to play an important, yet poorly understood, role in determining genome organisation and cell identity. The physical principles underlying the interplay between epigenetic dynamics and genome folding remain elusive. Here we propose and study a theory that assumes a coupling between epigenetic mark and genome densities, and which can be applied at the scale of the whole nucleus. We show that equilibrium models are not compatible with experiments and a qualitative agreement is recovered by accounting for non-equilibrium processes which can stabilise microphase separated epigenomic domains. We finally discuss the potential biophysical origin of these terms.Establishing distinct and inheritable cellular identities in different tissues is crucial to the existence of complex multi-cellular organisms. Because all cells contain the same DNA, cellular fate cannot be directed by genetic cues alone. Instead, tissue-specific 3D genome organisation [1] and biochemical (also called epigenetic) patterns [2] are thought to be key regulators of cellular fate [3]. Epigenetic patterns are composed of biochemical marks on DNA and histones -the proteins which package DNA into chromatin [4] -and can be deposited or removed by a range of specialised proteins [2] often recruited by complexes such as RNA polymerase [5] and Polycomb Repressive Complex (PRC) [6].Investigating the interplay between genome organisation and epigenetic patterns can shed light into the mechanism underlying cell fate decision [7,8]. To study the dynamics of epigenetic patterns, Ising-like models have been proposed in the biophysics literature [9][10][11][12][13][14]. Yet, the genome is poorly represented by a 1D array of Ising spins. Instead, it is a fluctuating polymer which can assume distinct spatial organisations in 3D and also in response to stimuli [15,16] and epigenetic cues [3].In this Letter, and the companion paper [17], we aim to analyse the link between chromatin large-scale organisation and epigenetic dynamics by using models for chromatin folding inspired by the physics of magnetic polymers [18]. We study a Landau-Ginzburg field theory where the dynamics of epigenetic marks is linked to that of genome folding within the nucleus. Our theory considers a 1D chain of Potts-like spins which is allowed to fluctuate in 3D. It markedly departs from previous works [19,20] as we formalise and study a theory for the epigenomic organisation of a whole eukaryotic nucleus and investigate how non-equilibrium processes affect its thermodynamics and kinetics.We show that while an equilibrium theory for epigenomic organisation captures some features seen in vivo, such as segregation of different epigenetic marks, it fails to explain the experimentally observed coexistence of diverse epigenetic and genomic domains in eukaryotic nuclei [2,21]. In light of...