  Homogenization is a powerful way of taming a class of finite structures with
several interesting applications in different areas, from Ramsey theory in
combinatorics to constraint satisfaction problems (CSPs) in computer science,
through (finite) model theory. A few sufficient conditions for a class of
finite structures to allow homogenization are known, and here we provide a
necessary condition. This lets us show that certain natural classes are not
homogenizable: 1) the class of locally consistent systems of linear equations
over the two-element field or any finite Abelian group, and 2) the class of
finite structures that forbid homomorphisms from a specific MSO-definable class
of structures of treewidth two. In combination with known results, the first
example shows that, up to pp-interpretability, the CSPs that are solvable by
local consistency methods are distinguished from the rest by the fact that
their classes of locally consistent instances are homogenizable. The second
example shows that, for MSO-definable classes of forbidden patterns, treewidth
one versus two is the dividing line to homogenizability.
