  We introduce the Brownian spatial coalescent, a class of Markov spatial
coalescent processes on the $d$-dimensional torus in continuous space, that is
axiomatically defined by the following property: conditional on the times and
locations of all coalescence events, lineages follow independent Brownian
bridges backwards in time along the branches of the coalescence tree. We prove
that a Brownian spatial coalescent is characterised by a set of finite
"transition measures" on the torus, in analogy to the transition rates that
characterise a non-spatial coalescent.
  We prove that a Brownian spatial coalescent is sampling consistent, in a
novel sense, if and only if all transition measures are constant multiples of
Lebesgue measure, and the multiples are the transition rates of a
$\Xi$-coalescent, or a $\Lambda$-coalescent if simultaneous mergers are not
allowed. This defines the "Brownian spatial $\Xi$-coalescent", which we prove
describes the genealogies of the $\Xi$-Fleming-Viot process - a generalisation
of the well known Fleming-Viot process - at stationarity, and in fact describes
its full time reversal if augmented with "resampling" steps at the times of
coalescence events.
  An important consequence of our results is that all spatial population models
in which individuals follow independent Brownian motions and the branching
mechanism depends non-trivially on the spatial distribution, for example
through local regulation, have non-Markovian genealogies.
  Byproducts of our results include explicit formulas for samples from the
stationary distribution of a $\Xi$-Fleming-Viot process, Wright-Malecot type
formulas, and a representation of the backward dynamics of lineages in terms of
Brownian motions with coupled drift. This includes calculations of the drift
that leads to multiple or even simultaneous mergers in any dimension.
