To be honest that sounds like a mess of an environment to work on. I would be pushing back on whoever said retries are not acceptable by saying that the flaky environment you are expected to work on is a cause of X percentage of the flaky tests. By the way when you say 'prod team', do you mean the developers?
So as well as pushing back on their opinion. You can improve your tests by designing your test approach following all the sections listed in Cypress recommendations of Core Concepts:
https://docs.cypress.io/guides/core-concepts/introduction-to-cypress
* random network errors - Do you wait for troublesome requests to finish?
* DOM failures - Do the manual testers have to deal with this too? Does the production environment experience this too?
* test code logic - Improve your test code by implementing all the core concepts linked above
* apply logic - what does this mean?
* temporary outages on external dependencies etc - consider stubbing external dependencies if possible.
The only alternative to retries is working on a properly functioning test environment and writing strong test code. Also, can you test on a different environment? Meaning if it's a DEV or QA environment that's a mess, can you test on a pre-production Stage environment instead?
All these questions are something for you to bring back to your team.