The Coulomb phase, with its dipolar correlations and pinch-point-scattering
patterns, is central to discussions of geometrically frustrated systems, from
water ice to binary and mixed-valence alloys, as well as numerous examples of
frustrated magnets. The emergent Coulomb phase of lattice-based systems has
been associated with divergence-free fields and the absence of long-range
order. Here, we go beyond this paradigm, demonstrating that a Coulomb phase can
emerge naturally as a persistent fluctuating background in an otherwise ordered
system. To explain this behavior, we introduce the concept of the fragmentation
of the field of magnetic moments into two parts, one giving rise to a magnetic
monopole crystal, the other a magnetic fluid with all the characteristics of an
emergent Coulomb phase. Our theory is backed up by numerical simulations, and
we discuss its importance with regard to the interpretation of a number of
experimental results
