We will introduce the notions of virtual conditions and balanced conditions in definable partial orders, and apply these ideas to forcing extensions of the Solovay model. We will present a variety of independence results, and some ZF implications, for weak forms of the Axiom of Choice, and discuss related issues concerning the preservation of cardinals and chromatic numbers. Some of these results can be stated more generally using a natural classification of definable partial orders in terms of amalgamation properties.