Achieving international food security requires improved understanding of how international trade networks connect countries around the world through the import-export flows of food commodities. The properties of international food trade networks are still poorly documented, especially from a multi-network perspective. In particular, nothing is known about the multi-network's community structure. Here we find that the individual crop-specific layers of the multi-network have densely connected trading groups, a consistent characteristic over the period 2001-2011. Further, the multi-network is characterized by low variability over this period but with substantial heterogeneity across layers in each year. In particular, the layers are mostly assortative: more-intensively connected countries tend to import from and export to countries that are themselves more connected. We also fit econometric models to identify social, economic and geographic factors explaining the probability that any two countries are co-present in the same community. Our estimates indicate that the probability of country pairs belonging to the same food trade community depends more on geopolitical and economic factors-such as geographical proximity and trade-agreement co-membership-than on country economic size and/or income. These community-structure findings of the multi-network are especially valuable for efforts to understand past and emerging dynamics in the global food system, especially those that examine potential 'shocks' to global food trade.