Do these people believe they are actually physical descendents of Ephraim, or that they are grafted into Israel? The second is 2H and biblical.
Grafted is a good point of focus.
And logially, there are some factors to consider that make a world of difference. Many claim that the judgement of the Lost Tribes are somehow an action of Finality......and though I agree with him on how the tribes themselves are also found within the NT, I don't think it'd be far off to say that the 10 Tribes were also scattered to other nations. I find it interesting to see how often it seems that there's resistance to the thought that it may be impossible to think that somehow the Jews could have somehow been centered at ONE location rather than being spread abroad all over the earth.....especially when seeing the many ways that Jewish groups/cultural aspects have been found all over the world---be it in Ethopia or India and many other places.
Of course, it seems that many will automatically shut down the moment anything is mentioned of the term "Two House Theory"...but for myself, I think it'd be reasonable to say that many of those deemed to be "Gentile" nations/cultures are indeed descended or connected with Israelite culture in one way or another....while at the same time its true that there are still Gentiles who never had contact with Israelitie culture and yet they were still used of the Lord. It doesn't necessarily have to be an "either or scenario of saying that either all Gentiles are the descendants of Israel (Ephraim/Manasseh and others) or that there are no Gentiles whom God never used to bless Israel.
As the Jews were literally scattered ALL OVer the world in many Diasporas/eras, my mindset is that it should not be surprising to find Hebrew camps/traces of Israelite culture in many parts of the world---for in many ways, it can be seen as an insurance policy enacted by God to make certain His people were all over and within the nations themselves so that they'd either influence for the sake of Divine Protection...or learn something from the nations that they needed to surive.
I'm one who believes God is revealing His Jewish/Gentile Remnant all around the world and supporting "Two-House Theory" of the Gentiles being of the House of Joseph/Ephraim that was scattered (Genesis 48:19), with God bringing his people home in reunification prior to the Messiah’s return (Isaiah 11:12-16; Jeremiah 10:6-10; Ezekiel 37:15-28; Zechariah 10:6-10, Romans 11, Romans 15, etc). Its not difficult for me to consider the House of Ephraim being connected to Gentiles in light of how Joseph Himself was Egyptian in many ways when saving his people--from shaving his beard to taking on an Egyptian Name and marrying an Egyptian wife......and it makes sense that we must understand that the Gentile nations Jacob spoke of in Genesis 48:19 are rooted in both Joseph an Israelite and his Egyptian wife, with these Gentile nations connected to Israel through their father Joseph. In some respects Joseph is also a type of Messiah.
Unlike many Two-House Theory advocates, I think that the dispersed do show up in scripture as not being lost as a whole. For the New Testament mentions Anna from the tribe of Asher (Luke 2:36-37) and James was written to the twelve tribes dispersed (not lost) in James 1:1-2. Peter writes to them as well (I Peter 1:1-2)---and thus, on this part I believe that the proponents of such "Two House" Theory are exaggerating at best the fact of the tribes being "lost" if thinking they're no longer present ANYWHERE after the exile. However, to be fair, I also believe that it is entirely possible, if not very plausible, that there were some, if not many of those 10 tribes that were disbursed that did lose their identity (probably on purpose) to fit in better with society.
Many of them (if not most) already had no love for their ancestors or the God of their ancestors. Technically, it would only take ONE generation of even one small number of the 10 tribes to do this to turn into 10's of millions of people that are truly Israelites that are still scattered throughout all of the earth, essentially being Gentiles. It'd be foolish for one to suppose that EVERY SINGLE ISRAELITE that was dispersed maintained their identity, as if there was NEVER a teenager that got so tired of being harassed that they decided to change his or her name to become like their neighbor. So the Two House Movement may be a little to "extreme", if you will, in many aspects...but any honest student of the word certainly has to admit that it is EXTREMELY possible that there WAS some families within the tribes that dissented and by default would create millions of "lost tribes" today.
Some may say, "But what about Romans 11 and the issue of the Gentiles being Grafted into Israel?"...and this is something many think shows that Gentiles never had any dealings with the Jewish people. On that, I can agree..and on the issue, I want to be clear that I'm NOWHERE NEAR SUPPORTING the falsehood that all things "Gentile" are to be ashamed of if its not connected to what's Jewish....for when it comes to God's Workings in the world, it seems clear from scripture that God used both JEWS and GENTILES not connected with anything Jewish to do mighty things. God loves the Gentiles disconected from Jewish culture just as much as He loves the Hebrews......and its not an issue of who's "SUPERIOR" since both groups have been used by God.
Essentially, if you're a Gentile, Work it out and be proud of your heritage
However, with that said, I also think that it is possible for the term "Gentile" to have more than one meaning...and it may be wrong to assume all references to "Gentiles" is somehow seperate from Jewish culture.
In example, I'm reminded of what God had said in Romans 9. For seven hundred years before Jesus's birth, Hosea told of God's intention to bring Gentiles into His family after the Jews rejected his plan (Hosea 2:23 and Hosea 1:10).
Romans 11 describes the Gentiles as Wild Olive Branches who've been grafted on the unproductive tree that is the Jews...the Jews being like natural branches of that tree (Romans 11:11-30). Gentile believers were grafted into the tree like a wild olive branch to inject new vitality into the old tree..and both Jews and Gentiles share its nourishment and depend on Christ for life while neither cna rest on heritage, culture, or theological beliefs for salvation.
But on the issue, how does a branch become wild? It seems that it was already the case in scripture that there were believers in God outside of the nation of Israel/Abraham (i.e. Melchizedek, Jethro, Job, Noah, The Roman Centurion, Cornelius, Naaman the Syrian, etc) whom the Lord used...and thus, it would seem (IMHO) a stretch to say that Romans 11 was about God saying he was not going to begin using the Gentiles who had no worth before or ability to be used by Him.
And with wild olive branches, if I'm not mistaken, in some cases, one can have a plant that's domesticated placed into the wild ....only to come back months or years later and find out that the plant was able to adapt to the elements and effectively become "wild." Kudzu comes to mind, as it is a serious invasive plant in the United States that been spreading in the southern U.S. at the rate of 150,000 acres (61,000 ha) annually. It was introduced from Japan into the United States in 1876 at the Philadelphia Centennial Exposition, where it was promoted as a forage crop and an ornamental plant. From 1935 to the early 1950s, the Soil Conservation Service encouraged farmers in the southeastern United States to plant kudzu to reduce soil erosion...and the Civilian Conservation Corps planted it widely for many years. It was subsequently discovered that the southeastern US has near-perfect conditions for kudzu to grow out of control — hot, humid summers, frequent rainfall, temperate winters with few hard freezes (kudzu cannot tolerate low freezing temperatures that bring the frost line down through its entire root system, a rare occurrence in this region), and no natural predators. As such, the once-promoted plant was named a pest weed by the United States Department of Agriculture in 1953. Kudzu is now common throughout most of the southeastern United States, and has been found as far northeast as Paterson, New Jersey, in 30 Illinois counties including as far north as Evanston, and as far south as Key West, Florida.
Its now all over the trees/landscape and in some cases choking life out of plants....whereas in other cases, there has been a symbosis that has occurred. The same principle seems to happen with animals who become feral over a period of time, even though once they were tame (i.e. Dingos in Austraila, Feral cats, etc)....and the same happens in reverse with animals who became domesticated
Thus, it'd seem possible that Gentiles in Romans 11 could have previously been connected with the Northern Kingdom....those of Israel who became "wild", in accordance with Genesis 48:19...and thus, what God was doing was truly bringing about a reunification of Jew/Gentile with the Gentiles having no room to boast.