An important property of Kingman's coalescent is that, starting from a state
with an infinite number of blocks, over any positive time horizon, it
transitions into an almost surely finite number of blocks. This is known as
`coming down from infinity'. Moreover, of the many different (exchangeable)
stochastic coalescent models, Kingman's coalescent is the `fastest' to come
down from infinity. In this article we study what happens when we counteract
this `fastest' coalescent with the action of an extreme form of fragmentation.
We augment Kingman's coalescent, where any two blocks merge at rate $c>0$, with
a fragmentation mechanism where each block fragments at constant rate,
$\lambda>0$, into it's constituent elements. We prove that there exists a phase
transition at $\lambda=c/2$, between regimes where the resulting `fast'
fragmentation-coalescence process is able to come down from infinity or not. In
the case that $\lambda