  Stratified-algebraic vector bundles on real algebraic varieties have many
desirable features of algebraic vector bundles but are more flexible. We give a
characterization of the compact real algebraic varieties having the following
property: There exists a positive integer r such that for any topological
vector bundle E on X, the direct sum of r copies of E is isomorphic to a
stratified-algebraic vector bundle. In particular, each compact real algebraic
variety of dimension at most 8 has this property. Our results are expressed in
terms of K-theory.
