In set-theoretic forcing, the construction of a model typically proceeds by iterating forcing notions along a linear order, each step adding new sets while preserving certain properties of the ground model. Shelah's homogeneous forcing seeks iterations rich in automorphisms --- symmetries of the forcing poset that permute the generic objects without altering the structure of the extension. The goal is to build models of ZF plus dependent choice at some cardinal kappa where every set, modulo a suitable ideal, is equivalent to a kappa-Borel set. The universe becomes, in a precise sense, tame.
The technical path requires three constraints on each forcing step: supports of size less than kappa, the kappa-plus chain condition, and strategic completeness below kappa. These ensure that the iteration does not collapse cardinals and that the partial order remains well-behaved at each stage. The automorphism requirement adds a fourth: the iteration along the index set must admit enough symmetries to make the resulting model sufficiently uniform.
But a critical subtlety emerges. The iteration of the index set gains many automorphisms, yet the forcing notion itself does not become homogeneous in the classical sense. The automorphisms permute the index coordinates but do not necessarily map names of generic reals onto each other. The symmetry is structural, not pointwise. The model has the uniformity property that was sought --- every set is Borel modulo the ideal --- but the tool used to build it is less symmetric than the thing it produces.
The through-claim is that symmetry in the output does not require symmetry in the process. A construction can produce a uniform result through means that are themselves only partially uniform. The product can be more symmetric than its factory.