The major problem is this bundle might not have a global section in general. This is where the "problem" of global anomalies comes in, for example. A global anomaly is different from the Gribov ambiguity, which is when a gauge fixing doesn't work to fix a gauge uniquely, in a global anomaly, there is no consistent definition of the gauge field. A global anomaly is a barrier to defining a quantum gauge theory discovered by Witten in 1980.