The Mishnah in Gittin debates a situation of a slave (
Eved Cna'ani) of two masters that one of the masters frees him, this turns him into a half freed man (and Jewish) and half slave (and not Jewish?!).
The solution of the Mishnah is to force the other master to free him.
Why is he not freed automatically like in the next Mishna (Mishna 6)? : "With regard to one who sells his slave to a non-Jew or to someone outside Eretz Yisrael, the slave automatically goes free"
Mishnah Gittin 4:5
One who is half slave and half free-man, serves his master one day and [works for] himself one day. These are the words of Beit Hillel. Beit Shammai said to them: "You have repaired [the situation] for his master, but for himself you have not repaired it. To marry a maidservant is impossible [i.e., forbidden], for he is half-free. [To marry] a free-woman is impossible, for he is half-slave. And was not the world created for the sake of reproduction, as it says (Isaiah 45:18) "Not for emptiness did He create it, but for settlement He formed it." Rather, due to Tikkun HaOlam, we force his master and he makes him a free-man, and [the slave] writes a document [of debt] for half his value. Beit Hillel retracted and ruled in accordance with the words of Beit Shammai.