The base class of PDL::Char is PDL; PDL::Core is a (very big) utility module that provides the pure-Perl methods and functions that enable use PDL etc to work.

Also, these days (as of PDL 2.048) the overloads (in the package PDL) happen in the modules that actually implement those functions: PDL::Ops for nearly everything, PDL::Primitive for x (matrix-multiply), and PDL::Core for the pure-Perl stringify.