  Baxter permutations, plane bipolar orientations, and a specific family of
walks in the non-negative quadrant are well-known to be related to each other
through several bijections. We introduce a further new family of discrete
objects, called coalescent-walk processes, that are fundamental for our
results. We relate these new objects with the other previously mentioned
families introducing some new bijections. We prove joint Benjamini--Schramm
convergence (both in the annealed and quenched sense) for uniform objects in
the four families. Furthermore, we explicitly construct a new fractal random
measure of the unit square, called the coalescent Baxter permuton and we show
that it is the scaling limit (in the permuton sense) of uniform Baxter
permutations. To prove the latter result, we study the scaling limit of the
associated random coalescent-walk processes. We show that they converge in law
to a continuous random coalescent-walk process encoded by a perturbed version
of the Tanaka stochastic differential equation. This result has connections (to
be explored in future projects) with the results of Gwynne, Holden, Sun (2016)
on scaling limits (in the Peanosphere topology) of plane bipolar
triangulations. We further prove some results that relate the limiting objects
of the four families to each other, both in the local and scaling limit case.
