Twenty years of genetic studies of marine invaders have shown that successful invaders are often characterized by native and introduced populations displaying similar levels of genetic diversity. This pattern is presumably due to high propagule pressure and repeated introductions. The opposite pattern is reported in this study of the brown seaweed, Sargassum muticum, an emblematic species for circumglobal invasions. Albeit demonstrating polymorphism in the native range, microsatellites failed to detect any genetic variation over 1,269 individuals sampled from 46 locations over the Pacific–Atlantic introduction range. Single‐nucleotide polymorphisms (SNPs) obtained from ddRAD sequencing revealed some genetic variation, but confirmed severe founder events in both the Pacific and Atlantic introduction ranges. Our study thus exemplifies the need for extreme caution in interpreting neutral genetic diversity as a proxy for invasive potential. Our results confirm a previously hypothesized transoceanic secondary introduction from NE Pacific to Europe. However, the SNP panel unexpectedly revealed two additional distinct genetic origins of introductions. Also, conversely to scenarios based on historical records, southern rather than northern NE Pacific populations could have seeded most of the European populations. Finally, the most recently introduced populations showed the lowest selfing rates, suggesting higher levels of recombination might be beneficial at the early stage of the introduction process (i.e., facilitating evolutionary novelties), whereas uniparental reproduction might be favored later in sustainably established populations (i.e., sustaining local adaptation).