On definable Galois groups and the strong canonical base property

In [E. Hrushovski, D. Palacin and A. Pillay, On the canonical base property, Selecta Math. (N.S.) 19(4) (2013) 865-877], Hrushovski and the authors proved, in a certain finite rank environment, that rigidity of definable Galois groups implies that T has the canonical base property in a strong form; "internality to" being replaced by "algebraicity in". In the current paper, we give a reasonably robust definition of the "strong canonical base property" in a rather more general finite rank context than [ E. Hrushovski, D. Palacin and A. Pillay, On the canonical base property, Selecta Math. (N.S.) 19(4) (2013) 865-877], and prove its equivalence with rigidity of the relevant definable Galois groups. The new direction is an elaboration on the old result that 1-based groups are rigid.