We consider invasion percolation on the complete graph ${\mathit{K}}_{\mathit{n}}$, started from some number $\mathit{k}(\mathit{n})$ of distinct source vertices. The outcome of the process is a forest consisting of $\mathit{k}(\mathit{n})$ trees, each containing exactly one source. Let ${\mathit{M}}_{\mathit{n}}$ be the size of the largest tree in this forest. Logan, Molloy and Pralat (2018) proved that if $\mathit{k}(\mathit{n})/{\mathit{n}}^{1/3}\to 0$ then ${\mathit{M}}_{\mathit{n}}/\mathit{n}\to 1$ in probability. In this paper, we prove a complementary result: if $\mathit{k}(\mathit{n})/{\mathit{n}}^{1/3}\to \infty$, then ${\mathit{M}}_{\mathit{n}}/\mathit{n}\to 0$ in probability. This establishes the existence of a phase transition in the structure of the invasion percolation forest around $\mathit{k}(\mathit{n})\asymp {\mathit{n}}^{1/3}$.

Our arguments rely on the connection between invasion percolation and critical percolation, and on a coupling between multisource invasion percolation with differently-sized source sets. A substantial part of the proof is devoted to showing that, with high probability, a certain fragmentation process on large random binary trees leaves no components of macroscopic size.