How do you coach a team on making user stories more independent?

I overheard a Dev and Tester talking about a user story where the Tester sent feedback to the Dev. The Tester said he didn’t feel comfortable passing the story because the way the acceptance criteria are written, he needs to test one other user story at the same time and pass both of them together.

This set off alarm bells in my head and I took the opportunity to mention that we might have issues with our stories not being independent.

I know about the acronym INVEST, but I feel like this is one of those skills that takes lots of practice and lots of reinforcement over time, so in general, how do you coach your teams to make their stories more independent? Are there any other acronyms or metaphors or “life stories” you use to help the team understand the concept and the importance? Can anyone share any success stories from their own experience?