  We analyse finite-time singularities of the Teichm\"uller harmonic map flow
-- a natural gradient flow of the harmonic map energy -- and find a canonical
way of flowing beyond them in order to construct global solutions in full
generality. Moreover, we prove a no-loss-of-topology result at finite time,
which completes the proof that this flow decomposes an arbitrary map into a
collection of branched minimal immersions connected by curves.
