A tagline is a punchline on top of a punchline. It’s that simple. To go back to our conversation analogy it’s when you’re friend makes a crack and then you quickly reply with your own joke that plays off his. You’re using his punchline as your setup.
Here’s an example from Mitch Hedberg:
“You know when it comes to racism people say ‘I don’t care if they’re black, white, purple or green’. Hold on now. Purple or green? You gotta draw the line somewhere… the hell with purple people… Unless they’re suffocating… then help ’em”
There are multiple punchlines in this example: “You gotta draw the line somewhere”, “The hell with purple people”, and “Unless they’re suffocating” are all punchlines. But the last two punchlines are taglines because they use the last punchline as their setup. If you were to take the last tagline “Unless their suffocating” and replace the original punchline of “You gotta draw the line somewhere” then the joke wouldn’t make since. It’s because taglines are not substitutes for punchlines… they work off them.
Don’t think that just because an audience is laughing means you need to start an entirely new joke. You can see from the example that Hedberg got several high quality jokes from the same setup. This allowed him to get more, high quality laughs per minute than he would have been able to if he started a new joke after the punchline “You gotta draw the line somewhere.”