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Ecological relationships between bacteria mediate the services that gut microbiomes provide to their hosts. Knowing the overall direction and strength of these relationships within hosts, and their generalizability across hosts, is essential to learn how microbial ecology scales up to affect microbiome assembly, dynamics, and host health. Here we gain insight into these patterns by inferring thousands of correlations in bacterial abundance between pairs of gut microbiome taxa from extensive time series data, consisting of 5,534 microbiome profiles from 56 wild baboon hosts over a 13-year period. We model these time series using a statistically robust, multinomial logistic-normal modeling framework and test the degree to which bacterial abundance correlations are consistent across hosts (i.e., "univeral") or individualized to each host. We also compare these patterns to two publicly available human data sets. We find that baboon gut microbial relationships are largely universal: correlation patterns within each baboon host reflect a mixture of idiosyncratic and shared patterns, but the shared pattern dominates by almost 2-fold. Surprisingly, the strongest and most consistently correlated bacterial pairs across hosts were overwhelmingly positively correlated and typically belonged to the same family - a 3-fold enrichment compared to pairs drawn from the data set as a whole. The bias towards universal, positive bacterial correlations was also apparent in monthly samples from human infants, and bacterial families that had universal relationships in baboons also tended to be universal in human infants. Together, our results advance our understanding of the relationships that shape gut microbial ecosystems, with implications for microbiome personalization, community assembly and stability, and the feasibility of microbiome interventions to improve host health.