I will discuss some results which are part of a preprint with Jaoui and Jimenez (2020), connecting with an earlier paper with Palacin (2017) and an even earlier one with Hrushovski andPalacin (2013). A strong version of the canonical base property (strong CBP) was formulated in the 2017 paper. We give some examples and counterexamples, discussing in particular a Galois theory of projective bundles(in complex geometry).