The land use sector of agriculture, forestry, and other land use (AFOLU) plays a central role in ambitious climate change mitigation efforts. Yet, mitigation policies in agriculture may be in conflict with food security related targets. Using a global agro-economic model, we analyze the impacts on food prices under mitigation policies targeting either incentives for producers (e.g., through taxes) or consumer preferences (e.g., through education programs). Despite having a similar reduction potential of 43-44% in 2100, the two types of policy instruments result in opposite outcomes for food prices. Incentive-based mitigation, such as protecting carbon-rich forests or adopting low-emission production techniques, increase land scarcity and production costs and thereby food prices. Preference-based mitigation, such as reduced household waste or lower consumption of animal-based products, decreases land scarcity, prevents emissions leakage, and concentrates production on the most productive sites and consequently lowers food prices. Whereas agricultural emissions are further abated in the combination of these mitigation measures, the synergy of strategies fails to substantially lower food prices. Additionally, we demonstrate that the efficiency of agricultural emission abatement is stable across a range of greenhouse-gas (GHG) tax levels, while resulting food prices exhibit a disproportionally larger spread.