I think unofficial factions are far more likely and easy to be formed than official political parties, at least in regards to Federation-wide policy.
In the
A Time to... series, the one thing that stood out for me is that, especially in the post-Dominion War/pre-
Destiny Federation, most Federation Councillors are going to be looking out for their constituent planet. For example, (and I apologize if I spoil anything), the issue arises of Federation aid to a planet on the Federation-Klingon border that had a Saddam Hussein-like dictator with advanced weapons supplied to him secretly by the Federation during the War, and who aggressively attempted to annex a Klingon colony world. Long story short (cutting out the details and skipping the facts that are unimportant to my example), the Andorian, Tellarite, and Betazoid Councillors arrive at the Federation President's office to inform him that they're going to introduce a bill cutting off aid to this planet and instead redirecting efforts to Betazed (at least, that's their public argument). The Federation President then mentions the deteriorating humanitarian conditions on the former enemy planet, and that he was
sure he wouldn't see anything in the bill dealing with Councillor's pet issues like upgrades to planetary transporter networks and so forth, and basically says that he'll veto that bill.
Another example is, in
Articles of the Federation, when the same Tellarite and Betazoid Councillors, among other members of an
ad-hoc coalition, threaten to vote to cut off aid to Cardassia for similar reasons, although in that case the Tellarite Councillor was doing it to get back at the President and other members of the Council for outvoting him on a nomination to the Federation Security Council. There the President's staff manage to negotiate and convince the politicians to not do that because of noble, humanitarian reasons and the like.
I'm probably not doing a good enough job with my examples, but I wanted to use them to show how and why temporary coalitions would be more likely to arise. Another example, and I forget which novel, says that the Councillor from Alpha Centauri went along with the majority almost all the time, provided it didn't directly affect or involve Alpha Centauri.
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Rush, the controversy you're referring to is in regards to the Federation trading with the planet Aligar, where 90% of the population is used in slave labor to mine for the material kellinite. The controversy was that the President basically questioned their patriotism and so forth, and thereby drastically slowed down consideration of the President's appointments or policies. I thought it was an interesting situation, when compared to real life politics. Another thing I liked was Bacco's "Travelgate"

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