My wife is constantly coughing and the medication she is on isn't working. It's a wet cough but the last doctor did not prescribe anti biotics. What must we do next. She is on a Multi vitamin, nose spray and antihistamine?

Revisit or. second opinion are needed. Not all coughs require antibiotics: 80++% of upper respiratory infections are viral. More importantly is cough control and several excellent Rxs are available. Until seen again use Delsym DB every 13 hrs if not allergic to this medicine.
Cough investigation. Requires review in investigation and management. Your practitioner may investigate with chest X-ray, sputum analysis and blood tests to assess underlying cause.