Approximately 30%-40% of global CO fixation occurs inside a non-membrane-bound organelle called the pyrenoid, which is found within the chloroplasts of most eukaryotic algae. The pyrenoid matrix is densely packed with the CO-fixing enzyme Rubisco and is thought to be a crystalline or amorphous solid. Here, we show that the pyrenoid matrix of the unicellular alga Chlamydomonas reinhardtii is not crystalline but behaves as a liquid that dissolves and condenses during cell division. Furthermore, we show that new pyrenoids are formed both by fission and de novo assembly. Our modeling predicts the existence of a "magic number" effect associated with special, highly stable heterocomplexes that influences phase separation in liquid-like organelles. This view of the pyrenoid matrix as a phase-separated compartment provides a paradigm for understanding its structure, biogenesis, and regulation. More broadly, our findings expand our understanding of the principles that govern the architecture and inheritance of liquid-like organelles.