The thing that actually shifted things for us was stopping the referee role entirely and making them solve it themselves first. When they'd come complaining about who gets the remote, instead of deciding for them, I'd say "you two figure out a system and come tell me what you decided." Most of the time they'd work it out in two minutes because there's no audience and no parent decision to argue against. It sounds counterintuitive when you're exhausted by the noise, but the second you step back from being the judge, a lot of the performance aspect of the fighting disappears.
That said, the earlier point about boredom versus actual conflict matters. With an 8 and 11 year old, there's a real gap in what they want to do, so constant proximity breeds tension. But instead of separating them (which just delays things), I found giving them a shared project actually works better - something that requires cooperation but isn't forced friendship. Building something together, working toward a goal, or even just having one activity they both genuinely enjoy reduces the low-level friction that turns into bickering. They don't have to love each other's company, but when they're working toward something, the dynamic shifts.
One more thing: timing matters. My kids get worse when they're tired or hungry, so if evening is peak bickering time, that's worth looking at. Sometimes what feels like a personality clash is just their nervous systems being fried. A snack, some outside time, or just lower expectations during that window can prevent half the arguments before they start.