  We consider a random walk on the first quadrant of the square lattice, whose
increment law is, roughly speaking, homogeneous along a finite number of
half-lines near each of the two boundaries, and hence essentially specified by
finitely-many transition laws near each boundary, together with an interior
transition law that applies at sufficient distance from both boundaries. Under
mild assumptions, in the (most subtle) setting in which the mean drift in the
interior is zero, we classify recurrence and transience and provide power-law
bounds on tails of passage times; the classification depends on the interior
covariance matrix, the (finitely many) drifts near the boundaries, and
stationary distributions derived from two one-dimensional Markov chains
associated to each of the two boundaries. As an application, we consider
reflected random walks related to multidimensional variants of the Lindley
process, for which the recurrence question was studied recently by Peign\'e and
Woess (Ann. Appl. Probab., vol. 31, 2021) using different methods, but for
which no previous quantitative results on passage-times appear to be known.
