Even within the same political party there's fragmentation.
International
Is Julian Assange a selfless defender of basic human rights, or a Bond-villainesque manipulator and possible rapist?
United States
The Republican Party: mostly social conservatives vs. the fiscal conservatives who really don't care about social issues. On top of that, are the Republicans a "big tent" party? Or are they shunning a lot of non-heartland WASP males at the cost of creating a bigger base? (In other words, are the party leaders unwittingly, or even intentionally breaking their own base?) The hiring of Michael Steele has brought this explosive topic to the forefront in the Republican party.
The fiscal conservatives (and libertarians) generally support gay rights, limited gun control, and sometimes abortion; these positions quite predictably infuriate the social conservatives. The fiscal conservatives, for their part, desperately wish that the social conservatives would just Stop Being Stereotypical.
Then there's paleo vs. neo: the conflict between paleo-cons vs neo-cons is basically about foreign policy. To put things stereotypically, the paleo-cons generally favor an extremely isolationist foreign policy while the neo-cons favor an extremely interventionist foreign policy.
John McCain. Basically Republicans' opinions of McCain tend to depend on their stance on social issues: 1. Republicans who are moderate-to-liberal on social issues tend to view McCain as a reasonable man who could have prevented the GOP's current perceived slide into the hands of the evangelical right had he won, and as a poor unfortunate Woobie who was ultimately a victim of George W. Bush's unpopular legacy (this faction points out that McCain had run against Bush in 2000 and was more moderate than him) and ended up getting unfairly saddled with an unfounded reputation as "Bush, Part II," and/or that his big mistake that doomed him was picking Sarah Palin as his running mate. 2. Meanwhile, Republicans who are socially conservative and evangelical tend to view Senator McCain as a sniveling "traitor" simply because he was relatively moderate on social issues, and claim that Sarah Palin was the ticket's only saving grace. The social conservatives also attempt to rationalize the 2008 results by claiming that it was McCain who lost the election and not Palin. Needless to say, the Republicans who still like John McCain generally have come to dislike Palin, while those who always disliked McCain tend to be diehard fans of Palin, which leads us to...
Sarah Palin: Put it this way: opinions on Palin with the Republican Party tend to divide in the following way: Social conservatives love her and believe that she's been slandered by the media, while fiscally conservative Republicans and libertarians tend to view her as an embarrassment to the party.
The Tea Party, and even within the tea party there's some dissension. Some feel that the grassroots movement was hijacked by Republicans. Many have noted that the Tea Party was started by the fiscal conservative and libertarian factions of the GOP base...but that it was quickly hijacked by the social conservatives.
Birtherism: most Republicans are moving away from it and criticizing other Republicans for using it, Claiming it to be a distraction from real issues.
Ron Paul seems to be a Base Breaker among Republicans nowadays too. At CPAC and other party gatherings, he seems to get a wildly enthusiastic reception from many people....but a rather cool or even hostile reception from everyone else. In large part, many of the people who hate Ron Paul seem to dislike him mainly because of the overzealousness of his hardcore supporters, rather than anything said or done by Paul himself.
The Republican "civil war" that has kicked into high gear since the 2008 election seemed to finally abate somewhat in late 2010/early 2011, only to burst wide open all over again with the beginning of the 2012 presidential race. The fiscal conservatives and social conservatives are at war once more. The only potential candidates who appeared to be able to unite all factions are Chris Christie, Marco Rubio, and Mitch Daniels, all three of whom have explicitly declined to run for President in 2012. So it seems the GOP civil war will continue for at least another year until the primaries are over.
The Democrats: seems to be split between compromising moderates (or "blue dogs"/DINO) and the actual liberal/progressive members. Some feel that the Republicans chased off the moderates and now they're all under the big tent of the Democrats controlling the party's direction.
One question that's usually debated among Dems/liberals is; Are the Dems really a liberal party? Or are they trying their best to be Republican-lite? The answer depends on which Dem/liberal you ask.
Barack Obama: Did he live up to his promises, or is he an ineffectual centrist who squandered the opportunity he had with the majority in the Senate, etc.?
Did Republicans get more out of Obama's compromises then Democrats?
Whether or not the Dems are taking the black/hispanic/gay/female votes for granted. And by extension...
Whether or not Obama should address African American issues directly, and should the black community expect him to. Is Obama avoiding addressing the black community because he knows they're gonna vote for him anyway? Or is it because he's afraid of being labeled biased toward blacks by the political right and mainstream media? Or both?
There's also contention within the black community over this, with some black Hillary Clinton supporters and black independents saying "I Told You So" to the black Obama supporters. There's also those black Obama supporters who are still loyal to him vs. the ones whom are starting to be vocal against him. This can get very nasty on and off the internet. Even among black journalists, political analysts, etc. it gets dicey.
It gets really cynical and dark when some suggest that Obama's race has actually hindered him from being a effective leader because he's scared of the Scary Black Man image.
In The Sixties, the divide in the Democratic Party was between three groups: white liberals and minority voters who opposed segregation, white southerners who supported it, and blue-collar workers who didn't care either way and were more concerned with labor issues. The Civil Rights Movement, The Vietnam War and the counterculture tore this coalition apart, with the white southern voters feeling that the rest of the Democrats had betrayed the white race, while the (largely Catholic) workers felt that the Democrats had become a party of immorality and anti-Americanism. The Republicans capitalized on this divide to build a new coalition of social conservatives, militarists, and their longtime allies in the business community, leading to their political dominance in The Eighties.
Speaking of the above, Republicans and Democrats may perhaps be considered Broken Bases of the US. This is YMMV, honestly, considering how similar the two parties actually are, especially when compared to the vast multiparty and ideological messes Europe (and the rest of the world, for that matter) gets into.
There's also congress, are they misunderstood? or a legion of obstructionists? Btw their disapproval rating was a whopping 82% in 2011.
All of the above is why people want a third or even a fourth party to choose from. Republican dissenters wanting a more moderate alternative to the current GOP whom they feel has a problem with Jumping Off the Slippery Slope into fringe territory, and liberals wanting a progressive alternative to what they see as the Dem's becoming republican-lite.
In this US, almost every single political issue deeply divides the country due to their nature of a two party system and heavy focus of mud slinging in virtually every debate.
Canada
Canadian politics, too. Most of the time, the war is held between the Liberals, the NDP, and the Conservatives. But throw in the Bloc Quebecois, and the Green Party, and you have a recipe for broken base politics. The only thing that people can reasonably agree upon is that the Green Party is too far out there to be given a seat.
Separatism may be quite literally a broken base issue, since it literally involves breaking up the country (though even the term "breaking up" is controversial to those who favor Quebec's absence). And now that group has a broken base itself with the majority of the youth in Quebec NOT wanting to separate from Canada*
as they really like Canada as a whole, and understand exactly what they get from being part of Canada and just how much they'd loose if they left
and undercutting the separatist's position and power to practically nothing.
Alberta (with addition to up and coming work in Saskatchewan) and many of the oil and gas projects break Canadian bases. On one hand, they're a major stimulator of the economy and are paying for many luxeries by the entire country. However, they also what causes Canada to snuggle a little too closely to the US and possibly China in the future. It gets even more muddled when possible environmental effects are taken into consideration.
United Kingdom
The Liberal Party at the end of World War I. Some, including the Prime Minister David Lloyd George, wanted to maintain the wartime coalition with the Conservatives, but others, including the party leader Herbert Asquith, wanted to split. Eventually the Conservatives broke away in 1922, and in the following election Lloyd George's National Liberals and Asquith's Liberals performed poorly. The Liberals never recovered from this and were overtaken by Labour as the party of the Left.
The Liberal Democrats were practically worshipped by many young adults, particularly because they promised to abolish university tuition fees. A hung parliament between the Conservatives and Labour followed; the Conservatives formed a coalition with the Lib Dems, who then went back on their promise on tuition fees (which, ironically, have subsequently been raised to a maximum of £9000/yr from the previous maximum of about £3200/yr). Not only did this cause a lot of division within the Lib Dems, with several of their members publicly expressing outrage, but also those who voted for them in the election.
First Past the Post or Alternative Vote?
The Labour Party - has it adapted to the times (for example, deindustrialisation) by becoming New Labour under Tony Blair, or has it betrayed the working classes by abolishing Clause 4 of the Labour party mainfesto, which stated that common ownership was a goal of the party? Those who say that Labour's policy post 1997 was simply "Thatcherism with a human face" would point out that many of Labour's policies during the Tony Blair administration do not fit with the traditional socialist ideology of the party, in particular expanding privatisation to many areas that even the Thatcher administration did not try to privatise. Those who argue against would point to the continued high spending on public services.
Historically, divisions in the 1980s in the Labour Party. The party as a whole tended to move towards the left in the early 80s with Michael Foot, which led eventually to a split in the party as the more right wing elements formed the Social Democratic Party in 1981 (who later merged with the Liberals).
The Conservative Party and opinions on the European Union. Current division within the party is between the "soft" Eurosceptic opinion of David Cameron who supports repatriation of powers and "hard" Eurosceptic opinion that support withdrawal from the EU. There was a debate in Parliament over having a referendum on Europe where many Tory M Ps voted against the party to approve the idea of a referendum.
This was a particularly destructive issue with the Major government of 1990-97. Major was a soft Europhile and adopted a stance of integration with Europe which proved unpopular with much of the party. There was much talk of the party ousting Major so he resigned as party leader in 1995 to initiate a leadership election. While he won, it didn't save his government who lost in the 1997 General Election to Labour.
In Scotland sepcifically, there is a huge divide over nationalisation or unionism. Some,epecially the Scottish National Party, want Scotland to exist as an independant country, with complete control at Holyrood in Edinburgh, rather than Westminister in London, over 100 miles away. Others think its a ridiculous and unnecessary idea, and sticking with the UK is the only sensible option, because Scotland is too small to support itself economically, and anyway there have been no major problems with the Union in over 300 years. A third party wants more devolved powers for Scotland,while still remaining part of the United Kingdom.
Australia
The Australian Labor Party recently. Kevin Rudd was leader of the party and prime minister from 2007 to 2010. Then Julia Gillard came along, kicked him out of the top job and appointed herself, based on a vote by their colleagues. Labor supporters (and everyone else) are split - was it horrible bullying behaviour or smart politics?
The Australian Democrats suffered from this more than most parties, as there are/were members and voters who were loyal to the visions of each particular leader. Particularly divisive were:
Cheryl Kernot - did she betray her party to join the ALP or was she just being a sensible politician?
Meg Lees - did she collude with the Howard government to bring in the GST, or was she used by savvier political operators?
Natasha Stott-Despoja - was she the party's last best hope of relevance, or was she a power-mad bitch queen?
State leaders sometimes seem more divisive than national ones in Australia: Joh Bjelke-Peterson (Queensland) and Jeff Kennett (Victoria) are particularly notable for how polarising they each were.
If you think about it the vast majority of Australian politics can be seen as this trope.
