I think the paper "[Inverses of structured vector bundles](https://doi.org/10.1007/s10711-015-0081-9 "Biswas, I., Pingali, V.P. Inverses of structured vector bundles. Geom Dedicata 179, 279–285 (2015)")" has exactly what you need.


  [1]: https://link.springer.com/article/10.1007/s10711-015-0081-9