We present a comprehensive experimental and theoretical study of the kagome ice Coulomb phase that explores the fine-tuning of critical correlations by applied field, temperature, and crystal orientation. The continuous modification of algebraic correlations is observed by polarized neutron scattering experiments and is found to be well described by numerical simulations of an idealized model. We further clarify the thermodynamics of field-tuned Kasteleyn transitions and demonstrate some dramatic finite-size-scaling properties that depend on how topological string defects wind around the system boundaries. We conclude that kagome ice is a remarkable example of a critical and topological state in a real system that may be subject to fine experimental control.