  We explain in depth the previously proposed theory of the coherent Van der
Waals(cVdW) interaction - the counterpart of Van der Waals (VdW) force -
emerging in spatially coherently fluctuating electromagnetic fields. We show
that cVdW driven matter is dominated by many body interactions, which are
significantly stronger than those found in standard Van der Waals (VdW)
systems. Remarkably, the leading 2- and 3-body interactions are of the same
order with respect to the distance $(\propto R^{-6})$, in contrast to the
usually weak VdW 3-body effects ($\propto R^{-9}$). From a microscopic theory
we show that the anisotropic cVdW many body interactions drive the formation of
low-dimensional structures such as chains, membranes and vesicles with very
unusual, non-local properties. In particular, cVdW chains display a
logarithmically growing stiffness with the chain length, while cVdW membranes
have a bending modulus growing linearly with their size. We argue that the cVdW
anisotropic many body forces cause local cohesion but also a negative effective
"surface tension". We conclude by deriving the equation of state for cVdW
materials and propose new experiments to test the theory, in particular the
unusual 3-body nature of cVdW.
