There are ways to separate yourself from your sibling, but unless for a good reason it’s a pretty crappy thing to do from the clans’ perspective. In their eyes StarClan placed you and your sibling(s) together for a reason and breaking that bond isn’t right (flawed logic of course, but one the clans continue to believe, though they’re getting better at it as the years go on). Most siblings also don’t want to separate. Conjoined siblings are greatly attuned to one another and it’s very rare for them to feel the need to separate. There’s also repercussions for this, cats lose strength, stamina, and are very weak following the separation. Not to mention their shared thoughts are quieted. It takes a medicine cat to use their connection to StarClan to break it. Tawny would most likely ask Tigerstar to use Littlecloud to break it so she could join Shadowclan since Bramble refused. He would be her best bet, fat chance anyone in Thunderclan would help her leave. Bramble was pretty devastated for a while. He was already battling with his self worth and now his sibling wanted to leave him? But he ultimately wanted Tawny happy and not trapped to him if she wasn’t. They’d travel to the moonstone with Littlecloud and get separated, with Bramble returning to Thunderclan alone.