Invasive species are one of the main threats to global biodiversity and, within marine ecosystems, tunicates feature some prominent examples. Styela plicata is an ascidian species inhabiting harbours in all temperate oceans and seas, thus being considered a thriving invasive species. However, this species' adaptive mechanisms, introduction history, and population structure have never been completely elucidated. Here, by genotyping 87 S. plicata individuals from 18 localities worldwide with 2b‐RADseq, we confirm the global presence of four chromosome inversions, demonstrate population structuring on this species, detect local adaptation signals, and infer historical demographic events. We show that North Carolina individuals constitute an unrelated population, Atlanto‐Mediterranean and Pacific localities form their own genetic clusters with substructuring, being the most evident the split between northern and southern Atlantic localities. The locality of South Carolina presents an intermediate genetic position between North Carolina and the other two groups pointing to a hybrid origin with recurrent gene flow. We generate and test demographic models, providing evidence of two independent introduction events to the Atlantic and Pacific, and an admixture that originated the population of South Carolina. Finally, we identify candidate loci for adaptation, with functions involved with cell processes, metabolism, development, and ion transport, among others. Overall, this study highlights the complex historical processes of S. plicata, which have led this species to its current distribution, population structure, and local adaptation footprint in oceans worldwide.