  Interacting particle systems undergoing repeated mutation and selection steps
model genetic evolution, and also describe a broad class of sequential Monte
Carlo methods. The genealogical tree embedded into the system is important in
both applications. Under neutrality, when fitnesses of particles are
independent from those of their parents, rescaled genealogies are known to
converge to Kingman's coalescent. Recent work has established convergence under
non-neutrality, but only for finite-dimensional distributions. We prove weak
convergence of non-neutral genealogies on the space of c\`adl\`ag paths under
standard assumptions, enabling analysis of the whole genealogical tree.
