Genomic variation of an invasive species may be affected by complex demographic histories and evolutionary changes during biological invasions. We described the relative influence of bottlenecks, clonality, population expansion, and crossover interference in determining genomic variability of the widespread red macroalga Agarophyton vermiculophyllum. Its introduction from mainland Japan to the estuaries of North America and Europe coincided with shifts from predominantly sexual to partially clonal reproduction and rapid adaptive evolution. A survey of 62,285 SNPs for 351 individuals from 35 populations was aligned to 24 chromosome-length scaffolds. Our results reinforce the central importance of demographic history, and founder effects in particular, in driving genomic variation in invasive populations, even when localized adaptive evolution and reproductive system shifts are observed.