Bands that split for political reasons
Rule one of keeping the band together- don't mention the vote.
For many years we have been told not to bring up politics in polite company, for the reason that it way more often than not turns ugly. A beloved band breaking up because of the differing political beliefs of members should serve as a warning not only to its fans but to the rest of us that not everything needs politicising. While there is definitely a market for politically-charged music, you’re not going to get very far if the members all have different views.
Let's see some bands who broke up not because of who kept putting brown M&Ms in the sweet jar, but because of that *other* nonmusical pressure:

Public Enemy
On top of the rap game for over three decades, the legends broke up in 2020 over Flavor Flav's opposition to the band playing a Bernie Sanders rally in Los Angeles. The band later refuted this despite Flavor tweeting “Are you kidding me right now??? ... over Bernie Sanders??? You wanna destroy something we’ve built over 35 years OVER POLITICS???” According to Dorian Lynskey of The Guardian: “Flavor’s position in the band had survived not only a $2,600-a-day crack habit but also four seasons of the dating show Flavor of Love. However, this was the last straw.”Florida-Georgia Line
The popular US country duo called it quits in 2022 in part due to Brian Kelley's vocal support of former President Donald Trump. This didn't bode well with his partner Tyler Hubbard, who even went as far as to unfollow him on social media during the 2020 election.The Eagles
Not exactly a case of interband political conflict here, but still related to politics. Things had been going south for a long time before the 1980 benefit for LA politician Alan Cranston, but this is where things came to a head. Before the show, guitarist Don Felder clearly wasn’t thrilled to meet the US senator’s wife, giving her a "Nice to meet you . . . I guess." This didn’t go down well with frontman Glenn Frey, who spent most of the show threatening to “kick (Felder’s) ass”. “Just three more songs!” Felder avoided a beating by leaving the venue (and band) as soon as they played the final note, triggering their split and a declaration by Don Henley that The Eagles would reform “when hell freezes over”. Satan would be seen skating to work in 1994.Elegant Machinery
The Swedish synthpop band split for the second time in 2011, apparently due to band members expressing discomfort at keyboardist (and professional Alex Kapranos lookalike) Richard Jomshof’s move into very right-wing politics. Like with Public Enemy above, Jomshof denied that the split had a political element, claiming that “the rest of the band actually don't get along with (singer) Robert (Enforsen) anymore”, but it’s worth noting the rest of the band reformed without him five years later. Jomshof now serves as secretary of the conservative Sweden Democrats, while the band carries on as a duo after the departure of Leslie Bayne in 2023.Radio Werewolf
Probably the craziest band featured here, Radio Werewolf were LA goth poppers noted for a controversial aesthetic which combined satanic and Nazi imagery (sample lyric: “Eva, oh Eva (Braun), come sit on my face”). Regular guests on US talkshows during the height of the “satanic panic”, they were genuinely involved with the Church of Satan, but intended their use of Nazi aesthetics as dark humour. When actual neo-Nazis began turning up at their concerts, most of the original band bailed (drummer “Evil Wilhelm” told a German interviewer that “What began as a labour of love and a true manifestation of ‘gothic’… gradually became more and more misunderstood.”), and so frontman Nikolas Schreck and his wife Zeena LaVey (daughter of the founder of the Church of Satan) moved to Germany and continued the band until 1993.
It’s telling that most of the bands on this list are American. The US is very politically polarised, especially post-2016, and it is not uncommon for apolitical people to be called “indecisive” at best and “covering for extreme beliefs” at worst. In general, the most successful acts in music have been apolitical, believing that music should unite people of all beliefs, colours and creeds. But if you can’t keep a band together over political differences, how is that going to keep the fans together?