  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 toric Hermitian ample line bundles, we
give necessary and sufficient condition for existence of a sequence of point
configurations which is asymptotically Fekete (and hence possess good sampling
properties) with respect to each one of the line bundles. When they exist, we
also present a way of constructing such sequences. As a byproduct we get a new
equidistribution property for maximizers of products of Vandermonde
determinants.
