  The classical model for the genealogies of a neutrally evolving population in
a fixed environment is due to Kingman. Kingman's coalescent process, which
produces a binary tree, universally emerges from many microscopic models in
which the variance in the number of offspring is finite. It is understood that
power-law offspring distributions with infinite variance can result in a very
different type of coalescent structure with merging of more than two lineages.
Here we investigate the regime where the variance of the offspring distribution
is finite but comparable to the population size. This is achieved by studying a
model in which the log offspring sizes have a stretched exponential form. Such
offspring distributions are motivated by biology, where they emerge from a toy
model of growth in a heterogenous environment, but also mathematics and
statistical physics, where limit theorems and phase transitions for sums over
random exponentials have received considerable attention due to their
appearance in the partition function of Derrida's Random Energy Model (REM). We
find that the limit coalescent is a $\beta$-coalescent -- a previously studied
model emerging from evolutionary dynamics models with heavy-tailed offspring
distributions. We also discuss the connection to previous results on the REM.
