We characterize braided commutative Yetter-Drinfeld C *-algebras over weak Hopf C *-algebras in categorical terms. Using this, we then study quotient type coideal subalgebras of a given weak Hopf C *-algebra G and coideal subalgebras invariant with respect to the adjoint action of G. Finally, as an example, we explicitly describe quotient type coideal subalgebras of the weak Hopf C *-algebras associated with Tambara-Yamagami categories.