We introduce a process where a connected rooted multigraph evolves by splitting events on its vertices, occurring randomly in continuous time. When a vertex splits, its incoming edges are randomly assigned between its offspring and a Poisson random number of edges are added between them. The process is parametrised by a positive real $λ$ which governs the limiting average degree. We show that for each value of $λ$ there is a unique random connected rooted multigraph $M(λ)$ invariant under this evolution. As a consequence, starting from any finite graph $G$ the process will almost surely converge in distribution to $M(λ)$, which does not depend on $G$. We show that this limit has finite expected size. The same process naturally extends to one in which connectedness is not necessarily preserved, and we give a sharp threshold for connectedness of this version. This is an asynchronous version, which is more realistic from the real-world network point of view, of a process we studied in arXiv:1506.02697, arXiv:1703.09011.