Corrosion-resistant adhesive compositions are provided. The compositions are useful for bonding together plies in multi-ply cellulosic fibrous structures. Multi-ply cellulosic fibrous structures incorporating the adhesives, products produced from such structures, and methods of producing the structures are also disclosed. The adhesive compositions incorporate a polyvinyl alcohol polymer and a corrosion inhibitor. The adhesive compositions may be sprayed or coated onto the cellulosic structures to bond layers of the structure to each other. The corrosion-resistant adhesive compositions are useful for preventing corrosion in the various devices used for handling and application of the adhesives such as spray equipment and applicator rolls. The cellulosic fibrous structures may be used to produce a variety of products including paper towels, toilet tissue, facial tissue, napkins, and other absorbent materials.