  The forging of strong correlations on decreasing temperature can take place
without the arousal of conventional order. If this happens, as in some
geometrically frustrated magnets, disorder can be a phenomenon more interesting
than order itself. A Coulomb phase, for example, has critical-like pair-spin
correlations, leading to neutron scattering \textit{pinch points} and emergent
electromagnetism. Here we present a new instance of disorder in an Ising
pyrochlore lattice: the \textit{Polarized Monopole Liquid} (PML), a dense
monopole fluid with pinch points in the \textit{magnetic charge}-pair
correlations. It is a phase of "monopole matter" never considered before which,
in principle, can be stabilized in real materials using a magnetic field and
uniaxial stress along the [100] direction. To explain how the monopole
correlations arise, we show that the PML is a Coulomb phase in which spin
fluctuations cannot be assigned either to monopoles or to internal magnetic
moments, but necessarily comprehend both degrees of freedom. We develop a
simple but nontrivial method to Helmholtz decompose the spin field into a
divergenceless and a divergenceful part in magnetic charge disordered
pyrochlores that shows the appearance of pinch points associated to the
divergenceful component in places where Bragg peaks are observed for the
"all-in/all-out" antiferromagnet.
