A Death in the Family is the infamous 1988 storyline in which Jason Todd died at the hands of the Joker. The storyline closes with a confrontation between Joker and Batman in which Superman plays a key role; the story reveals the tension that DC was stressing at the time between the two, but it also reveals certain things haven't changed.
Superman enters the story after Jason's death. Batman is hunting down the Joker and has a lead that he needs to talk to the new Iranian ambassador. Superman shows up to inform him that the new ambassador (guess who) has diplomatic immunity and thus Batman cannot take him in.
At this point, by the way, Superman doesn't know that the Joker has killed Robin--Batman hasn't told him. In fact, canonically very few people in the DCU know anything about the second Robin's death, I believe.
The issue ends with Joker emerging from the limo and mocking Batman because he has diplomatic immunity.
Man, Clark and Bruce break my heart here. Bruce is so totally strung out and Clark is simply puzzled and worried...
The next issue starts with Clark and Bruce meeting with a CIA agent:
Although Clark's playing the heavy for the government, he drops the role when he and Bruce are alone together to talk seriously, and it's very sweet to see. Though, uh...Bruce needs to stop standing on ramps at random while arguing with Clark, lol.
The story continues with Bruce angsting about whether he has the right to kill the Joker and whether he's going too far for vengeance, but he shows up at the UN for a speech the Joker is about to give apparently ready to kill him.
Joker gives a long, rambling speech that becomes increasingly ominous, and yet not a soul in the crowd thinks it might be a good idea to vacate the premises. DCU citizens, don't ever change. Then Joker releases a bunch of Joker gas as we continue to follow Bruce's thoughts:
Yes, even Batman is impressed with Clark's ability to take things into his body. What? He said it, not me.
Moving on from the stupid innuendos, I love how sure Bruce is that Clark will be there and save the day. Superman returns his trust in the next panel:
Superman then flies off to disperse the gas in space. Putting aside how Clark is talking with his lungs full of Joker gas, keep in mind that he leaves the Joker with Batman, who has sworn to kill him. Either Clark trusts that he won't, or he has decided that he's willing to let Batman do it. There's an analogous scene in the "Public Enemies" arc of Superman/Batman from 2003 in which Batman tells Clark that if he wants to kill Lex, the two of them will cover it up and no one will ever know. At a certain level it shows that they have total faith the other hero will not kill. If you want to find a darker meaning in it, it's a hint that either is willing to let the other kill, given the right circumstances.
So by 1988, Superman and Batman are still fundamentally friends, although they are more abrasive than before (and some of that is due to Bruce's understandable grief and horror at Jason's death). But leading up to 1990, DC will push them further into barely-friends territory.
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