I always thought it was a misconception to be "half-Jewish". Usually the speaker means one of their parents are Jewish, making them either fully Jewish or fully Gentile.
However, I saw today that the Minchas Chinuch (Kometz HaMincha #347) suggests the possibility of such a status (although in Minchas Chinuch he seems less sure). The case is two people own a non Jewish slave (not an Eved Kenaani, but with קנין הגוף). One frees their half, and the half slave converts. The free half is Jewish, and when the remaining half is freed, it remains Gentile.
I'm looking for other sources which discuss such a status. Maybe they disagree with the concept and say the Jewishness spreads (as Minchas Chinuch suggests, unlike in his Kometz HaMincha). Maybe they discuss the practical ramifications that the Kometz HaMincha brings, or suggest new ones.
I'm already aware the Ben Ish Chai in Benayahu (Eruvin 18a), amongst other places, discusses this concept, and relates it to Siamese twins. The Kometz HaMincha also quotes a Turei Even which discuses this, but I don't know what he says.