  Free probability of type B was invented by Biane--Goodman--Nica, and then it
was generalized by Belinschi--Shlyakhtenko and F\'evrier--Nica to infinitesimal
free probability. The latter found its applications to eigenvalues of perturbed
random matrices in the work of Shlyakhtenko and C\'ebron--Dahlqvist--Gabriel.
This paper offers a new framework, called ``free probability of type
B${}^\prime$'', which appears in the large size limit of independent unitarily
invariant random matrices with perturbations. Our framework is related to
boolean, free, (anti)monotone, cyclic-(anti)monotone and conditionally free
independences. We then apply the new framework to the principal minor of
unitarily invariant random matrices, which leads to the definition of a
multivariate inverse Markov--Krein transform.
