Summary: Completing former works by the author [Markov Process. Relat. Fields 22, No. 2, 267–309 (2016; Zbl 1359.60060); ALEA, Lat. Am. J. Probab. Math. Stat. 14, No. 2, 751–798 (2017; Zbl 1372.60102); Electron. Commun. Probab. 24, Paper No. 47, 15 p. (2019; Zbl 1422.60061)], we study the recurrence of an inhomogeneous Markov chain in the plane, when the environment is horizontally stratified and the heterogeneity of quasi-periodic type.