  We show that the chiral kagome ice manifold exhibits an anomalous integer
quantum Hall effect (IQHE) when coupled to itinerant electrons. Although
electron-mediated interactions select a magnetically ordered ground state, the
full ice manifold can coexist with the IQHE over a range of finite
temperatures. The degenerate ice states provide a natural realization of
power-law correlated flux disorder, for which the spectral gap of the system
remains robust. The quantized (up to exponentially small finite-temperature
corrections) Hall conductance persists over a wide range of electron densities
due to the disorder-induced localization of electronic states.
