Here is a article on "Catholic Integralism".
"Since his nomination as the Republican candidate for vice president, focus has intensified on J.D. Vance?s religious beliefs and how they connect to his politics.

Vance is a convert to Catholicism and seems to have the same policy positions that many American Catholic conservatives hold: opposition to abortion, support for the traditional family, skepticism regarding liberal immigration policies and efforts to combat climate change, and advocacy of economic tariffs.

Some news reports have also referenced Vance?s apparent association with Catholic Integralism, although Vance himself has not addressed the issue publicly.

So, now might be a good time to ask: What is Catholic Integralism?

What is Catholic Integralism?
The basic position of Catholic Integralism is that there are two areas of human life: the spiritual and the temporal, or worldly. Catholic Integralists argue that the spiritual and temporal should be integrated ? with the spiritual being the dominant partner. This means that religious values, specifically Christian ones, should guide government policies.

Catholic Integralists disagree about how to achieve this integration between the spiritual and temporal. Some argue that Christians, particularly Catholics, should have advisory roles in government and lead by example. Other Catholic Integralists want a more comprehensive approach to organizing society along Christian principles.

Catholic Integralists share an opposition to liberalism. Generally, liberalism is understood as a political philosophy that supports limits on the government?s authority and constitutional protections for the rights of individuals and minorities. But Catholic Integralists argue that liberalism is incapable of establishing deep forms of human community because it values individualism and liberty above all things.

The irony, from the Catholic Integralist perspective, is that liberalism is not really that liberal. Instead, liberalism demands ? and enforces ? adherence to a certain set of values, like tolerance and pluralism, that prevents creating a social order in which human beings can realize a larger, God-given meaning for their lives."..https://rhodeislandcurrent.com/2024/07/22/gop-vp-nominee-j-d-vance-is-linked-to-catholic-integralism-what-is-it/

I never actually heard the word "Integralism" before, but the concept is definitely part of Catholic understanding. It goes right back to Constantine and church "fathers" like Augustine.
One baptist author used the word "sacral society". The idea of a "sacral society" is that a society cannot function when there is a wide diversity of beliefs and values. It needs to be a sacral society in which it's citizens are bound together by a common religious loyalty which is maintained with civil authority .

That was part of Constantine's motive in uniting Rome with Christianity. To have a united society bound together by a common religious loyalty.
When Constantine came into the Church, he did not lay aside his weapons or legions at the door. He entered the church with the understanding that he would be the final arbitrator and defender of this "unifying" sacral society. This put a new weapon in the hand of the Church. History celebrates Constantine's "conversion" as the end of persecution. No, it was NOT the end of persecution, it simply brought the steel sword into the Church for it's use.

The church father's made a doctrine to support the use of this "sword", they took the Bible text in Luke 22:34, "Lord, here are two swords," and reinterpreted that to create a teaching saying Jesus intended His Church to have two swords, one the "sword of the Spirit" which the clergy wields, and "the sword of steel" which the law enforcing soldier uses.

By 1150 AD this interpretation of the two swords was well ingrained. Two swords were said to belong to Peter, one, the religious sword is in his hand, the other (steel sword) is at his command whenever it is needful to draw it. This doctrine was put in use all through medieval history.

Even the Reformers had a hard time understanding that this "two swords" interpretation was WRONG and began to persecute in territories under protestant control. It wasn't until the late 1700's that the two were separated and the church could no longer use a civil sword to enforce their religion. Church and state were separated, to allow religious freedom.

This integralism of course is undoing the very foundation of religious freedom, for the person who believes a sacral society is needed for a country or territory to function in a proper manner will see the union of church and state as a necessary thing. It's giving the sword of civil power back into the hands of the church.