A search for a problem-free cosmology within the framework of an effective non-minimally coupled scalar-tensor theory is suggested. With an appropriate choice of couplings in variants of a Lee-Wick model (as also in a model supporting Q-ball solutions), non-topological solutions (NTSs), varying in size up to 10 kpc to 1 Mpc can exist. We explore the properties of a `toy' Milne model containing a distribution of NTS domains. The interior of these domains would be regions where effective gravitational effects would be indistinguishable from those expected in standard Einstein theory. For a large class of non-minimal coupling terms and the scalar effective potential, the effective cosmological constant vanishes identically. The model passes classical cosmological tests and we describe reasons to expect it to fare well as regards nucleosynthesis and structure formation.