One thing you can try doing is to rework sections of the codebase gradually over time as you change them. Don't touch code that is working and doesn't need to be updated, but if you are working on a feature in one corner of the codebase, see what you can do to improve that part of the codebase.

This is easier to deal with both in terms of selling it to management and making sure you don't actually break the whole system compared to a full scale rewrite. I've made that mistake before and it wasn't fun. As others have said, code that's in production and works and is making money has value. The "cleanness of the code" is not something business owners care about, unless it causes problems.