Synthesis, thermodynamic properties, and microscopic magnetic model of ilinskite-type KCu5O2(SeO3)2Cl3 built by corner-sharing Cu4 tetrahedra are reported, and relevant magnetostructural correlations are discussed. Quasi-one-dimensional magnetic behavior with the short-range order around 50 K is rationalized in terms of weakly coupled spin ladders (tubes) having a complex topology formed upon fragmentation of the tetrahedral network. This fragmentation is rooted in the non-trivial effect of the SeO3 groups that render the Cu-O-Cu superexchange strongly ferromagnetic even at bridging angles exceeding 110°.