Klebsiella species are able to colonize a wide range of environments and include worrisome nosocomial pathogens. Here, we sought to determine the abundance and infectivity of prophages of Klebsiella to understand how the interactions between induced prophages and bacteria affect population dynamics and evolution. We identified many prophages in the species, placing these taxa among the top 5% of the most polylysogenic bacteria. We selected 35 representative strains of the Klebsiella pneumoniae species complex to establish a network of induced phage–bacteria interactions. This revealed that many prophages are able to enter the lytic cycle, and subsequently kill or lysogenize closely related Klebsiella strains. Although 60% of the tested strains could produce phages that infect at least one other strain, the interaction network of all pairwise cross-infections is very sparse and mostly organized in modules corresponding to the strains’ capsule serotypes. Accordingly, capsule mutants remain uninfected showing that the capsule is a key factor for successful infections. Surprisingly, experiments in which bacteria are predated by their own prophages result in accelerated loss of the capsule. Our results show that phage infectiousness defines interaction modules between small subsets of phages and bacteria in function of capsule serotype. This limits the role of prophages as competitive weapons because they can infect very few strains of the species complex. This should also restrict phage-driven gene flow across the species. Finally, the accelerated loss of the capsule in bacteria being predated by their own phages, suggests that phages drive serotype switch in nature.