Take $P \rightarrow M$ to be a principal $G$-bundle. We modify the bundle so the structure group $G$ is not the same for all fibres of the bundle.
Is there a name for such an object? Are there issues with modifying a principal bundle so the structure group is not constant?
It seems like there might be issues with defining trivializations for such a bundle. If there are, can they be overcome?
Motivation
Write $G = \text{SU} \left( 2 \right)$. In his paper "Quantum Field Theory and the Jones Polynomial", Witten considers Chern-Simons theory using a $G$-bundle $P \rightarrow \Sigma$, where $\Sigma$ is a Riemann surface.
Next he considers when $\Sigma$ is punctured at points $z_i$. He argues that, to get the correct phase space using quantization, we must modify $P$ so that the structure group is $G$ for all fibres except at the puncture points $z_i$. Here he says the structure group must be reduced to $G/T$, where $T$ is a maximal torus of $G$.