  For the renormalised sums of the random $\pm 1$-colouring of the connected
components of $\mathbb Z$ generated by the coalescing renewal processes in the
"power law P\'olya's urn" of Hammond and Sheffield we prove functional
convergence towards fractional Brownian motion, closing a gap in the tightness
argument of their paper.
  In addition, in the regime of the strong renewal theorem we gain insights
into the coalescing renewal processes in the Hammond-Sheffield urn (such as the
asymptotic depth of most recent common ancestors) and are able to control the
coalescence probabilities of two, three and four individuals that are randomly
sampled from $[n]$. This allows us to obtain a new, conceptual proof of the
asymptotic Gaussianity (including the functional convergence) of the
renormalised sums of more general colourings, which can be seen as an
invariance principle beyond the main result of Hammond and Sheffield.
  In this proof, a key ingredient of independent interest is a sufficient
criterion for the asymptotic Gaussianity of the renormalised sums in randomly
coloured random partitions of $[n]$, based on Stein's method.
  Along the way we also prove a statement on the asymptotics of the coalescence
probabilities in the long-range seedbank model of Blath, Gonz\'alez Casanova,
Kurt, and Span\`o.
