Illegal Religion: Well, now. While neither the Fundamental Contract nor the Imperial Charter considers freedom of religion a fundamental right (those would all be much more, um, fundamental), it is, in most ways, a strict subset of those which it does recognize. The latter does, however, mention freedom of philosophy in the clause which establishes the state religion:

The above notwithstanding, the freedom of philosophy for the individual shall not be abridged, save when required for the public safety; and the rights of the citizen-shareholder shall not be diminished or enlarged on any philosophical criterion; save that the doctrines of a philosophy may act as an impairment to citizenship when they are considered antithetical to true allegiance or the principles upon which the Empire is founded, and the Senate and Curia have made such determination.

– the Imperial Charter, Section II, Article VIII

Thus, no religion is illegal in the Empire per se.

That said, the Empire is very keen on certain principles, like the ethical equality of all sophonts, their endowment with certain absolute, inalienable, non-derogable rights, that these rights are to life and property, liberty, and the pursuit of eudaimonia, the obligation of contracts, and so on and so forth, and if your religion, philosophy, or culture differs significantly on that point – especially but not necessarily in praxis¹ – that’s what’ll get it on the list of Proscribed Promulgators of Pernicious Irrationality.

But, y’know, it’s not a per se ban, it’s a because you are by your own choice and statement incapable of undertaking the obligations inherent in Imperial citizen-shareholdership ban, which the Senate and Curia will be kind enough to explain the details of to you in the Take Your Religion/Culture/Philosophy and Shove It Act (As Applicable).

(Life is, by and large, a little more pleasant for those civilized henotheists who have no problem coming to a polite and respectful accommodation between their private beliefs and the public – primarily Flamic – ones. Dogmatic monotheism isn’t illegal, mind – its practice just makes you look like a right dick.)

Footnotes:

1. With regard to the “The religion requires or encourages behavior that is unacceptable to the ruling culture. In this case, the rulers may tolerate abstract belief in the religion as long as the objectionable elements are not practiced.” policy mentioned on the trope page, the Empire has thought about it for all of a second and then dismissed it. Remember that old Minbari saying, “Understanding is not required, only obedience”²? The Imperials prefer to espouse the opposite – especially the acquiescents. All deeds grow from thoughts, after all.
2. Having that in the doctrines of your religion isn’t banworthy, but it’s certainly a bad sign.

While TV Tropes only asks for Rule of Cautious Editing Judgement where Real Life examples are concerned, I’m going to ask that if anyone has the urge to discuss in the comments, we keep it in the hypothetical mode. I’m sure we can all think of such examples one way or t’other, but I do not want Earthling religious flamewars here, and will be striking down on any that do appear with great vengeance and furious anger, m’kay?