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Coral and algal holobionts are assemblages of macroorganisms and microorganisms, including viruses, Bacteria, Archaea, protists and fungi. Despite a decade of research, it remains unclear whether these associations are spatial-temporally stable or species-specific. We hypothesized that conflicting interpretations of the data arise from high noise associated with sporadic microbial symbionts overwhelming signatures of stable holobiont members. To test this hypothesis, the bacterial communities associated with three coral species (Acropora rosaria, Acropora hyacinthus and Porites lutea) and two algal guilds (crustose coralline algae and turf algae) from 131 samples were analyzed using a novel statistical approach termed the Abundance-Ubiquity (AU) test. The AU test determines whether a given bacterial species would be present given additional sampling effort (that is, stable) versus those species that are sporadically associated with a sample. Using the AU test, we show that coral and algal holobionts have a high-diversity group of stable symbionts. Stable symbionts are not exclusive to one species of coral or algae. No single bacterial species was ubiquitously associated with one host, showing that there is not strict heredity of the microbiome. In addition to the stable symbionts, there was a low-diversity community of sporadic symbionts whose abundance varied widely across individual holobionts of the same species. Identification of these two symbiont communities supports the holobiont model and calls into question the hologenome theory of evolution.