  We study the averaged products of characteristic polynomials for the Gaussian
and Laguerre $\beta$-ensembles with external source, and prove Pearcey-type
phase transitions for particular full rank perturbations of source. The phases
are characterised by determining the explicit functional forms of the scaled
limits of the averaged products of characteristic polynomials, which are given
as certain multidimensional integrals, with dimension equal to the number of
products.
