We study K-stability for tensor products of diagonal AH-algebras with arbitrary C*-algebras. Our main result provides a characterization of K-stability: for a diagonal AH-algebra A=lim(Ai,φi)
Nearby in the stack
,
A⊗B
is
K
-stable for every C*-algebra
B
if and only if the sizes of the matrix blocks in the inductive system grow without bound. As applications, we show that non-
Z
-stable Villadsen algebras of the first kind are
K
-stable when tensored with any C*-algebra. Moreover, any simple, unital, infinite-dimensional diagonal AH-algebra automatically satisfies this growth condition, and therefore its tensor product with arbitrary C*-algebras is always