  Baxter permutations, plane bipolar orientations, and a specific family of
walks in the non-negative quadrant, called tandem walks, 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 and we relate it
to the three families mentioned above.
  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 random measure on the unit square, called the Baxter
permuton and we show that it is the scaling limit (in the permuton sense) of
uniform Baxter permutations. In addition, we relate the limiting objects of the
four families to each other, both in the local and scaling limit case.
  The scaling limit result is based on the convergence of the trajectories of
the coalescent-walk process to the coalescing flow -- in the terminology of Le
Jan and Raimond (2004) -- of a perturbed version of the Tanaka stochastic
differential equation. Our scaling result entails joint convergence of the
tandem walks of a plane bipolar orientation and its dual, extending the main
result of Gwynne, Holden, Sun (2016), and giving an alternative answer to
Conjecture 4.4 of Kenyon, Miller, Sheffield, Wilson (2019) compared to the one
of Gwynne, Holden, Sun (2016).
