Marine plankton populate 70% of Earth's surface, providing the energy that fuels ocean food webs and contributing to global biogeochemical cycles. Plankton communities are extremely diverse and geographically variable, and are overwhelmingly composed of low-abundance species. The role of this rare biosphere and its ecological underpinnings are however still unclear. Here, we analyse the extensive dataset generated by the Tara Oceans expedition for marine microbial eukaryotes (protists) and use an adaptive algorithm to explore how metabarcoding-based abundance distributions vary across plankton communities in the global ocean. We show that the decay in abundance of non-dominant operational taxonomic units, which comprise over 99% of local richness, is commonly governed by a power-law. Despite the high spatial turnover in species composition, the power-law exponent varies by less than 10% across locations and shows no biogeographical signature, but is weakly modulated by cell size. Such striking regularity suggests that the assembly of plankton communities in the dynamic and highly variable ocean environment is governed by large-scale ubiquitous processes. Understanding their origin and impact on plankton ecology will be important for evaluating the resilience of marine biodiversity in a changing ocean.