Increasing climate risk is making property insurance unaffordable and unavailable. I study a novel Australian policy response: government-provided, mandatory, risk-based reinsurance for cyclone damage in home insurance. Public reinsurance reduces premiums by 21\% and increases insurance availability by 11\%. These gains are not a subsidy but arise from eliminating large pre-existing markups in private reinsurance and catastrophe bond markets, flowing primarily to insurers most constrained by tail-risk exposure. The markup reduction stems from neutralizing the high premium for spatially-correlated and ambiguous risk, with increased competition providing additional benefits. This demonstrates that insurance market dysfunction originates from frictions in tail-risk reinsurance markets, and that targeted, cost-neutral interventions in these upstream markets can restore affordability and availability in home insurance.