The virome is one of the most variable components of the human gut microbiome. Within twin pairs, viromes have been shown to be similar for infants, but not for adults, indicating that as twins age and their environments and microbiomes diverge, so do their viromes. The degree to which the microbiome drives the vast virome diversity is unclear. Here, we examine the relationship between microbiome and virome diversity in 21 adult monozygotic twin pairs selected for high or low microbiome concordance. Viromes derived from virus-like particles are unique to each individual, are dominated by Caudovirales and Microviridae, and exhibit a small core that includes crAssphage. Microbiome-discordant twins display more dissimilar viromes compared to microbiome-concordant twins, and the richer the microbiomes, the richer the viromes. These patterns are driven by bacteriophages, not eukaryotic viruses. Collectively, these observations support a strong role of the microbiome in patterning for the virome.