Centrosomes are membraneless organelles that nucleate microtubules. At their core is a pair of centrioles that recruit pericentriolar material (PCM), a phase-separated condensate. In many cell types, including human cells, centrosomes are surrounded by endoplasmic reticulum-derived membranes of unknown structure and function. Using volume electron microscopy, we show that the C. elegans centrosome is surrounded by a membrane reticulum that we call the centriculum, for centr osome-associated membrane ret iculum . Increasing centriculum size by genetic means led to expansion of the PCM and increased microtubule nucleation capacity, an unexpected finding given that the PCM is a membraneless condensate. We provide evidence that the centriculum serves as a microtubule “filter” by limiting the number of microtubules that can elongate fully. We also show the centriculum fuses with the nuclear envelope during mitosis. We propose that this fusion contributes to nuclear envelope breakdown by transducing forces from the elongating spindle to the nuclear membranes.