We study the link between supermassive black hole growth and the stellar mass assembly of their host galaxies in the state-of-the-art Romulus suite of simulations. The cosmological simulations Romulus25 and RomulusC employ innovative recipes for the seeding, accretion, and dynamics of black holes in the field and cluster environments respectively. We find that the black hole accretion rate traces the star formation rate among star-forming galaxies. This result holds for stellar masses between 10 8 and 10 12 solar masses, with a very weak dependence on host halo mass or redshift. The inferred relation between accretion rate and star formation rate does not appear to depend on environment, as no difference is seen in the cluster/proto-cluster volume compared to the field. A model including the star formation rate, the black hole-to-stellar mass ratio, and the cold gas fraction can explain about 70 per cent of all variations in the black hole accretion rate among star forming galaxies. Finally, bearing in mind the limited volume and resolution of these cosmological simulations, we find no evidence for a connection between black hole growth and galaxy mergers, on any timescale and at any redshift. Black holes and their galaxies assemble in tandem in these simulations, regardless of the larger-scale intergalactic environment, suggesting that black hole growth simply follows star formation on galactic scales.