CRISPR-Cas systems provide bacteria with adaptive immunity against bacteriophages. However, DNA modification, the production of anti-CRISPR proteins and potentially other strategies enable phages to evade CRISPR-Cas. Here we discovered a Serratia jumbophage that evaded type I CRISPR-Cas systems, but was sensitive to type III immunity. Jumbophage infection resulted in a nucleus-like structure enclosed by a proteinaceous phage shell - a phenomenon only reported recently for distantly related Pseudomonas phages. All three native CRISPR-Cas complexes in Serratia - type I-E, I-F and III-A - were spatially excluded from the phage nucleus and phage DNA was not targeted. However, the type III-A system still arrested jumbophage infection by targeting phage RNA in the cytoplasm in a process requiring Cas7, Cas10 and an accessory nuclease. Type III, but not type I, systems frequently targeted nucleus-forming jumbophages that were identified in global viral sequence datasets. These findings explain why many bacteria harbour both RNA- and DNA-targeting CRISPR-Cas systems. Together, our results indicate that jumbophage nucleus-like compartments serve as a barrier to DNA-targeting, but not RNA-targeting defences, and that this phenomenon is widespread amongst jumbophages.