Dietary shifts are key for enhancing the sustainability of current food systems but need to account for potential economic, social and environmental indirect effects as well. By tracing physical quantities of biomass along supply chains in a global economic model, we investigate the benefits of adopting the EAT-Lancet diet and other social, economic and environmental spillovers in the wider economy. We find that decreased global food demand reduces global biomass production, food prices, trade, land use and food loss and waste but also reduces food affordability for low-income agricultural households. In sub-Saharan Africa, increased food demand and higher prices decrease food affordability also for non-agricultural households. Economic spillovers into non-food sectors limit agricultural land and greenhouse gas reductions as cheaper biomass is demanded more for non-food use. From an environmental perspective, economy-wide greenhouse gas emissions increase as lower global food demand at lower prices frees income subsequently spent on non-food items.