<p>A sequence of point configurations on a compact complex manifold is asymptotically Fekete if it is close to maximizing a sequence of Vandermonde determinants. These Vandermonde determinants are defined by tensor powers of a Hermitian ample line bundle and the point configurations in the sequence possess good sampling properties with respect to sections of the line bundle. In this paper, given a collection of Hermitian ample line bundles, we address the question of existence of a sequence of point configurations which is asymptotically Fekete with respect to each of the line bundles. We give a conjectural necessary and sufficient condition and prove this in the toric case.</p>