The main aim of this paper is to develop a general approach, which allows to extend the basics of Brudnyi-Kruglyak interpolation theory to the realm of quasi-Banach lattices. We prove that all K-monotone quasi-Banach lattices with respect to a L-convex quasi-Banach lattice couple have in fact a stronger property of the so-called K(p, q)-monotonicity for some $$0<q\le p\le 1$$ , which allows us to get their description by the real K-method. Moreover, we obtain a refined version of the K-divisibility property for Banach lattice couples and then prove an appropriate version of this property for L-convex quasi-Banach lattice couples. The results obtained are applied to refine interpolation properties of couples of sequence $$l^{p}$$ - and function $$L^{p}$$ -spaces, considered for the full range $$0<p<\infty $$ .