The TVTropes Trope Finder is where you can come to ask questions like "Do we have this one?" and "What's the trope about...?" Trying to rediscover a long lost show or other medium but need a little help? Head to Media Finder and try your luck there. Want to propose a new trope? You should be over at You Know, That Thing Where.
Generally, the ability for a hero to get away with collateral damage is Hero Insurance.
If it's a bait-and-switch sort of thing — "Ace, you made a big mess today, caused a lot of people a lot of headaches, you destroyed the hero base... but you annihilated those evil Aliens And Monsters and just saved us all. Good job, Ace." — then that's Arson, Murder, and Lifesaving.
I'm not sure why else someone would be kept around if there wasn't something good attached. At the very least, if it's a kind of "You destroyed the hero base and you're a total klutz, but you're The Chosen One so we just have to put up with your idiocy", then they're The Chosen Zero.
Edited by KD
Nah, it's more like they keep them around because they're like "you destroyed our base but since you're actually strong enough to do so we'll give you another chance and let you join us. we need someone with skills like yours on our side". but Arson, Murder, and Lifesaving is very close.
Edited by KingOfStickers
You might have considered these already, but Destructive Savior, Terrifying Rescuer, Nominal Hero?
Nope. The ace didn't rescue anyone. Their destructive actions are what caught the eyes of the heroes, and they want what is essentially a villain to join them. Roughly.
Not to come off as argumentative, but I have noticed a number of people doing this on Trope Finder lately — could you maybe narrow it down a little more in your initial query? If you could give the full example instead of piecing out a little at a time, you'll probably get better suggestions.
If you don't want to do that, have you looked at some of the related tropes for some of these? That might help us find what you're looking for, or if we don't have it. For instance, a dangerous or even villainous character who the heroes want to join them even though they're not heroic in their motivations at all, that is very much what Nominal Hero is all about. It's not specifically about being destructive, but it doesn't rule it out.
Edited by Unsung
Yes. I'm very sure that I've come across the exact trope that I'm looking for before. Nominal Hero is extremely close but I'm almost certain that there is a more specific trope out there.
An ace is kept around because they did a very bad but cool thing. For example, an ace destroys the hero base, yet they keep them around because they're willing to help and also they could use their skills.