  Let G be a reductive group over an algebraically closed field k. Consider the
moduli space of stable principal G-bundles on a smooth projective curve C over
k. We give necessary and sufficient conditions for the existence of Poincar\'e
bundles over open subsets of this moduli space, and compute the orders of the
corresponding obstruction classes. This generalizes the previous results of
Newstead, Ramanan and Balaji-Biswas-Nagaraj-Newstead to all reductive groups,
to all topological types of bundles, and also to all characteristics.
