This is your fundamental error.
There are not 2 peoples of God. John 17:22, Ephesians 4:4-6, +
With the Twelve Tribes of Israel, surely you know that the Jews represent only the 2 tribes of Judah and Bejamin? There were and still is many from the other tribes and Gentiles, who have converted to Judaism and become Jews.
But God's secret is the whereabouts and who are today the 10 Northern tribes. They were exiled for a decreed period, Ezekiel 4:4-6, and it was them who Jesus came to save. Matthew 15:24
He was successful and we Christians are the result. We Christians ARE the people of God, by faith and by descent!
I don't think Christians represent the Ten Tribes. Ezekiel the prophet speaks about the unification of the two houses. Consider Ezekiel 37:15-23 Here God indicates the unification of the two houses using an illustration. Before I get to that, I want to relate a bit of background.
As you might recall, at the schism of Shechem, the 10 northern tribes seceded from Israel because Rehoboam overtaxed the people. 1 Kings 12. Jeroboam returned from exile in Egypt in order to plead with Rehoboam to reduce the tax load. Not only did Rehoboam refuse, he decided to place additional taxes on the people. The schism of Shechem can be understood as a conflict between two men: Rehoboam who wanted to become more feared than his father, and Jeroboam, who wanted to live free in his own land.
In any case, upon hearing the word of Rehoboam, Jeroboam declared independence from the house of Rehoboam and established a house for himself. Jeroboam decided to build a new capital city and he established two other places, other than Jerusalem, for Israel to worship. Jeroboam built Shechem in the land of Ephraim. Given this background, we take a quick look at Ezekiel 37.
God illustrates the unification of the two houses with sticks. He tells Ezekiel to take one stick to represent one of the houses and another stick to represent the other house.* Ezekiel is to hold the two sticks together in his hand. The meaning of the illustration seems clear. God is going to bring the two houses together and hold them together by his power.
God refers to one of the sticks in this way. "‘For Joseph, the stick of Ephraim and all the house of Israel, his companions." Ephraim is mentioned because, as it says in 1 Kings, "Jeroboam built Shechem in the land of Ephraim." And before the unification,
Ezekiel 37:20-23
The sticks on which you write will be in your hand before their eyes. Say to them, ‘Thus says the Lord God, “Behold, I will take the sons of Israel from among the nations where they have gone, and I will gather them from every side and bring them into their own land; and I will make them one nation in the land, on the mountains of Israel; and one king will be king for all of them; and they will no longer be two nations and no longer be divided into two kingdoms. They will no longer defile themselves with their idols, or with their detestable things, or with any of their transgressions; but I will deliver them from all their dwelling places in which they have sinned, and will cleanse them. And they will be My people, and I will be their God.
This doesn't describe Christians at all. Christians weren't scattered among the nations; they became Christians in their respective nations. Christians were never part of the Davidic dynasty. They never defiled themselves with idols or with detestable things. The two sticks prophecy is understood in the context of the
schism of Shechem. A nation was split into two houses, and God promises to reunite the two houses again under King David.
I believe that Jesus will rule over them during the Millennial period.
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* House here means "dynasty"