Unfortunately this ideology already exists in OTL and it's called Theonomy (and/or Dominionism in context of American Protestantism).Here's another ideology I got bored and devised. It's based on what I think a Biblical literalist theocracy would look like, and it isn't a pleasant state (and one that would offend many Christians from every side of the political spectrum).
Salvationism
Salvationism is a Christian theocratic ideology that believes the purpose of the government is to lead people to Jesus Christ through establishing Biblical principles as the law of society. It arose out of a certain evangelical revival movement that encourages its followers to prepare the world for Jesus's return. Its members gradually subverted the government before taking over in a somewhat violent revolution.
Salvationism is a Protestant movement led by a nameless denomination []--this denomination is by far the most prominent. However, practical matters forced the government to institute limited ecumenicalism--churches which have signed a statement of doctrine are permitted to be state churches of the "Salvationist Fellowship". In practice, these are all similar evangelical and fundamentalist churches, with permitted deviations in doctrine primarily for worship style (i.e. forbidding instruments in church music) or matters of church governance. Other Christian churches are subject to varying degrees or harassment--Catholicism and Orthodox churches are banned, as are newer Christian movements such as the LDS Church, Christian Science, and Jehovahs Witnesses, but liberal or mainline Protestants, as well as fundamentalists who do not agree with the Salvationist doctrine, are generally allowed to continue worshipping provided their pastors do not preach doctrine contrary to the state.
There is no freedom of religion. Banned churches and non-Christian religions are not permitted to assemble for worship. Religious clothing such as the hijab or Sikh turban are strictly outlawed. Atheism, agnosticism, and irreligion are deemed blasphemous and practicioners imprisoned. The worst punishment is reserved for idol worshippers such as Hindus, Buddhists, and neopagans--they are executed by stoning. A controversy arose after a leading Catholic archbishop was stoned to death for "worshipping an idol of Mary"--the Salvationists decreed Catholic and Orthodox worship not idolatry (but still outlawed), yet extrajudicial punishment of Catholics is still common. Any proselytism of non permitted religions is strictly illegal. Religious literature of these faiths is banned. Judaism is banned for being adversarial to Jesus, but Jews who convert to Salvationism are permitted and encouraged to organise as Messianic Jews.
The government is centered around local churches, which themselves are grouped into regional churches, and then into the national church. Most all regional or national politicians are ordained deacons and ministers. A council of 7 men termed "Regional Church Elders" governs all regions, while the nation is ruled by 7 men termed "National Church Elders". By statute, the nameless denomination fills 2/3 of all open government posts, including 5 out of 7 posts on all regional and national councils. Government is partially elected, partially not--all local positions are elected, as are elections to the Council of Elders at the regional and national level, but nearly all other offices are indirectly elected or appointed. Only the "man of the household" may vote, and he must be an active member of a Salvationist church who can only vote in elections pertaining to his denomination (i.e. a Baptist may not vote in an election involving the nameless denomination). There is no women's suffrage at all, while adult men still living with their parents or elderly men being cared for by their adult children may only vote if the man of the household transfers his status to them.
Most controversially, Salvationism believes everyone is to hold property in common as the early church in Acts did. For practical matters, this means all property is church property and therefore the Salvationist Fellowship controls the economy, allocating resources as necessary. Business occurs only at the behest of the state. The economy can roughly be said to be state capitalist. Economic discrimination is rampant against perceived enemies of the state, and the government frequently intervenes in private business to ensure the atmosphere of the business is Christian, as is its means of doing business. Tithing is a major source of government income--to not tithe at least 10% of your income (or even more if one is deemed as being "especially blessed") without a valid reason is grounds for losing church membership, which is effectively the suspension of one's civil rights.
The government takes an active role in culture. Practically all secular culture is outlawed or restricted as being immoral. Alcohol remains legal, but highly taxed and restricted--bars are illegal, and drunkenness even in private can result in losing church membership. All other drug use is criminalised (although medical marijuana is legal, but strictly regulated). Secular music, films, etc. are permitted to exist, but must pass a stringent board of censors. Artists and directors who fail to pass the censors too many times are generally blacklisted, and some genres are defacto forbidden out of similarity to immoral sorts of music. Morality is strictly enforced in public--unmarried men and women are kept separate (most places are gender segregated or have separate seating areas for men and women), modest clothing is required at all times, public displays of affection are not tolerated. While the government does not officially sponsor them, private morality police exist in all jurisdictions and are usually deputised by local police departments to reduce their workload. Religious culture is generally permitted as long as it is doctrinally correct--Salvationist notoriously raided art museums to confiscate and destroy many Renaissance paintings for portraying the Bible in "too Catholic" a manner or otherwise "glorifying paganism".
There are numerous limitations on civil rights. There is no freedom of speech, and even public profanity may result in fines for un-Christian behavior. Anything LGBT+ related is banned, and those known to have partaken in homosexual sex are executed. Abortion is strictly illegal, and those receiving or performing an abortion are also stoned due to the government equating it with sacrifices to Moloch. Fornication and adultery are also capital crimes, where both parties to the act may also be executed. Most crimes are executed by stoning--the "stoning machines" installed in execution squares are infamous symbols of the Salvationist regime. Crimes may be punished by flogging, amputation, imprisonment, or forced restitution. The latter can sometimes result in the criminal being forced to serve as an indentured servant, losing their civil rights in the process. Both Old Testament and New Testament law are referred to for how society is governed. Prisons double as indoctrination camps. Those who are sentenced to prison for crimes are forced to listen to hours and hours of sermons by "prison ministers" and memorise Bible verses related to their crime; physical torture is common, and state ideology freely melds with Biblical doctrine in these sessions.
Notably, the government believes in the equality of all races and ethnic groups. Racism is deemed as contrary to the goals of bringing the world the Gospel of Jesus, and strict laws against hate speech are applied. However, these laws are also extended to those who fight against alleged structural inequalities between races and ethnic groups should they argue too forcefully. Antisemitism is distinguished between racial and religious--the former is illegal, the latter is government policy.
Salvationists actively promote creation science as the core of their policy on science. Any indication that evolution is true or that the world and universe are billions of years old is punished, and there was even a case where a biologist was stoned to death for asserting mankind evolved. This has impeded research in many fields and made advancing science difficult at best, although it is widely known that scientists often write in code and frequently read otherwise forbidden scientific journals. However, in other fields the Salvationists are still highly advanced and active in developing new methods to assure their dominion over the world.
Salvationist foreign policy is centered around world domination for the sake of saving souls. Branches of the Salvationist Fellowship operate in numerous countries, and the government actively propogates mission work. Their embassies function as churches and centers for coordinating mission work, which in some countries has resulted in their expulsion. Nearly all diplomatic staff are ordained ministers who have served as missionaries in the past. The International Salvationist Fellowship is a global council of churches that fancies itself the world government, but unlike the United Nations or Comintern is known for playing an active hand in member states. It is hoped that one day, there will be no borders and the International Salvationist Fellowship will be the sole government on the planet until the return of Jesus.
As a result, Salvationist foreign policy lacks pragmatism. Their chief international ally is Israel due to their belief that Israel is necessary for the end times, but there is great distrust between the Israel and the Salvationists due to the religious antisemitism of the latter and the increasing prominence of religious Zionism in the former. The Shin Bet closely monitor the Salvationist Fellowship branch of Messianic Jews in Israel due to suspecting them of plotting a coup. Besides Israel, most nations ally with Salvationists do so out of fear and economic opportunity, hoping Salvationist investment and imports will aid them in their own national goals.
The Salvationist military fancies itself as warriors for Christ. Only members of churches in the Salvationist Fellowship may join, and indoctrination occurs at all levels thanks to the very large Chaplain Corps, which in addition to their typical role, also function akin to commissars in communist regimes. The military is known for their involvement in public policing and has occasionally committed war crimes on the basis of Old Testament law, but to their credit, the Salvationists have tried to eliminate that tendency in the military in the name of Jesus's message of redemption. However, it is still expected that foreign prisoners of war will be denied the right to practice their religion and may suffer execution for idolatry.
[1] - This practice is akin to some denominations who use the name "Christian Church" or "Church of Christ", but I'd rather avoid the confusion.
This one seems very hard to get instituted in any country given Protestant countries are also the ones where fundamentalist Christians are very much against the idea of abolishing private property. I think you'd need a 19th century POD where some segment of the labour movement fuses with the early fundamentalist movement and spreads itself through religious revivals.
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