Clean the broken soap dish pieces with trisodium phosphate to get rid of any soap residue. Rinse well and dry thoroughly.
Apply a thin line of a two-part epoxy adhesive along one side of the broken tile edge. Leave room on the outside edge for the second line of adhesive.
Draw a thin line with a fast-drying adhesive along the outside edge of the soap dish. Cyanoacrylate is a commonly found fast-drying adhesive.
Press the broken pieces into place. Align the pieces quickly because the cyanoacrylate dries in seconds.
Leave the soap dish to cure for at least 24 hours. Consult your epoxy glue instructions for exact curing times. Do not use the shower until it the glue is cured.
Wipe up any excess glue using acetone nail polish remover on a cotton swab.