Goyim mixed in with the Jewish People

i learnt that when moshiach comes moshiach will point out to many people who had always assumed they are goyim just as their parents and grandparents and earlier assumingly were. moshiach will tell them you are jewish! join our nation in the redemption. these will be people whose ancestors 500 or 1000 (or whenever) years ago were forcibly converted to other religions or somehow else got mixed into the goyishe nations. (the ten lost tribes fit in this narrative?)

anyways, my curiosity pertains to the opposite parallel. goyim who got mixed into the jewish people. this could’ve happen in different ways. a child brought up by a jew but never converted (and all her descendants). someone who underwent a conversion that was for whatever reason invalid and not a good conversion. or even someone who pretended to be a jew, told his children they are jewish and that lie perpetuated for generations. there may be other ways this could’ve happened (reform/karaite type converts that later mixed into orthodox/rabbinate communities without further conversion). and when these type of situations occurred 500 or 1000 or longer years ago, you now have tens or hundreds of thousands of people, descending from the female lineage of these people, who think they are jewish but are not. these people may be all around us. (is this what people call the eruv rav? or maybe similar to the cusim.)

so when moshiach comes will he point them out and say you aren’t jewish? they cannot participate in the redemption, etc. presumably since they acted jewish all their lives they also fulfilled the seven noachide laws and thus qualified for righteous gentile status.