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We demonstrate that the low temperature ($T$) properties of a class ofanisotropic spin $S=1$ kagome (planar pyrochlore) antiferromagnets on afield-induced $\frac{1}{3}$-magnetization ($\frac{1}{2}$-magnetization) plateauare described by a model of fully-packed dimers and loops on the honeycomb(square) lattice, with a temperature-dependent relative fugacity $w(T)$ for thedimers. The fully-packed O(1) loop model ($w=0$) and the fully-packed dimermodel ($w=\infty$) limits of this dimer-loop model are found to be separated bya phase transition at a finite and nonzero critical fugacity $w_c$, withinteresting consequences for the spin correlations of the frustrated magnet.The $w>w_c$ phase has short loops and spin correlations dominated by power-lawcolumnar order (with subdominant dipolar correlations), while the $w<w_c$ phasehas dominant dipolar spin correlations and long loops governed by a power-lawdistribution of loop sizes. Away from $w_c$, both phases are described by along-wavelength Gaussian effective action for a scalar height field thatrepresents the coarse-grained electrostatic potential of fluctuating dipoles.The destruction of power-law columnar spin order below $w_c$ is driven by anunusual {\em flux fractionalization} mechanism, topological in character butquite distinct from the usual Kosterlitz-Thouless mechanism for suchtransitions: Fractional electric fluxes which are bound into integer values for$w>w_c$, proliferate in the $w<w_c$ phase and destroy power-law columnar order.

Flux fractionalization transition in anisotropic $S=1$ antiferromagnets and dimer-loop models