  Consider a population evolving as a critical continuous-time Galton-Watson
(GW) tree. Conditional on the population surviving until a large time $T$,
sample $k$ individuals uniformly at random (without replacement) from amongst
those alive at time $T$. What is the genealogy of this sample of individuals?
In cases where the offspring distribution has finite variance, the
probabilistic properties of the joint ancestry of these $k$ particles are well
understood, as seen in \cite{HJR20, J19}. In the present article, we study the
joint ancestry of a sample of $k$ particles under the following regime: the
offspring distribution has mean $1$ (critical) and the tails of the offspring
distribution are \emph{heavy} in that $\alpha \in (1,2]$ is the supremum over
indices $\beta$ such that the $\beta^{\text{th}}$ moment is finite. We show
that for each $\alpha$, after rescaling time by $1/T$, there is a universal
stochastic process describing the joint coalescent structure of the $k$
distinct particles. The special case $\alpha = 2$ generalises the known case of
sampling from critical GW trees with finite variance where only pairwise
mergers are observed and the genealogical tree is, roughly speaking, some kind
of mixture of time-changed Kingman coalescents. The cases $\alpha \in (1,2)$
introduce new universal limiting partition-valued stochastic processes with
interesting probabilistic structures which have representations connected to
the Lauricella function and the Dirichlet distribution, and whose coalescent
structures exhibit multiple-mergers of family lines. Moreover, in the case
$\alpha \in (1,2)$, we show that the coalescent events of the ancestry of the
$k$ particles are associated with birth events that produce giant numbers of
offspring of the same order of magnitude as the entire population size, and we
compute the joint law of the ancestry together with the sizes of these giant
births.
