A lot of great answers focusing on the election system, but also, don’t forget to have the legislative process reward intransigence. As someone who was born in Belgium and now lives on the West Coast of the US, I think you really want built-in, intractable gridlock. (I’ll omit specific contemporary examples, but you want structural incentives that every party responds to, and a bad stable equilibrium, not a country where parties happen to be run by crazy people.) It should not only be possible for a minority party to stop anything from getting done, it should be impossible to agree on anything without agreeing on everything. Most important, the only way to win concessions in the eventual Grand Bargain must be total, unanimous intransigence. If individuals defect, the whole party gets rolled. That way, any member who compromises and defects really is selling out.
There should, however, be a robust way for all the normal, routine business to go on: the last year’s budget is automatically reauthorized if a new one does not pass, a caretaker government remains in place that must maintain the status quo, there are independent civil service or military officials with a free hand to deal with crises who are a lot more trusted and respected than politicians, a lot of policy is in the hands of other levels of government that are more functional. That way, there’s no overwhelming urgency and no outside pressure to force a bargain. It’s stable. People don’t feel all that bickering affects their daily lives or accomplishes anything, so they tune it out. It becomes a sport. If stuff only happens when party leaders agree, and election returns short of a massive landslide don’t change who has to agree or what they can agree on, it becomes just a game. For drama, though, there can be one looming dead-man switch which the country cannot just keep muddling through.
When nobody can ever agree, nothing is ever anybody’s fault. The cause of dysfunction today is the same as yesterday and tomorrow: the parties couldn’t reach a compromise. Saying who should have compromised is the same as saying whose position is wrong.
Make sure every faction listens exclusively to their own side’s propaganda and doesn’t live around anybody who openly disagrees with them. Even better is if they literally speak different languages. Give them a history that poisons the well, then make it so there can be no cooperation across either ethnic or ideological lines.
You can raise the pressure by having some hot-button issue that constitutionally cannot be solved by the legislature, so the activists displace all that frustration onto whatever they can disrupt.
Make sure politicians can’t trade horses: you can’t throw in some pork for someone’s hometown in exchange for a vote if there are no local districts and all contracts must be awarded to the lowest qualifying bid. You can’t trade a vote here for a vote there if only one omnibus bill can pass. These can even be a well-intended anti-corruption effort. Prevent backroom negotiations and make people say which of their positions they’d be willing to compromise on in public.
Be sure to have your party lists decided by the most committed activists, who show up to all the meetings, not corrupt party bosses. Corrupt bosses pick people who will return the favor by going along with the deals they negotiate.
It helps if people with no interest in a functioning democracy (an authoritarian monarch, foreigners, oligarchs) have a lot of influence over the process. This doesn’t have to be evil: looking at this country, are the separatists who want it to break up entirely or the radicals who think forcing a constitutional crisis is the only way forward totally wrong?