A multilayer adhesive film comprising surface adhesive layers, an internal Controlled Bond Layer (“CBL”) comprising a propylene-based polymer strippably adhered to an adjacent layer; an internal Strong Bond Layer (“SBL”) different than the CBL having a facial surface in adhering contact to the CBL and; optionally, one or more different internal Filler Layer(s) located between the SBL and the adhesive surface layer on the side of the SBL opposite to the adhered CBL. In preferred embodiments the adhesive comprises an ethylene/ethylenically unsaturated carboxylic acid copolymer. The adhesive films are used in cable shielding structures.