If Reuben's sister, Hanna, is a slave and she and Reuben have sexual intercourse, are their children slaves or mamzeirim?
Acc. to the Mishna in Yevamot 4:12-13, the offspring of a forbidden union, the penalty for which is karet, is a mamzer. That is the opinion of Rabbi Shimon haTeimani in mishna 13 - contra the opinions of Rabbi Akiva and Rabbi Yehoshua - and is understood to be the opinion of the sages (contra Akiva) in mishna 12.
Acc. to the Mishna in Ketubot 3:12, the offspring of such a forbidden union - one in which the woman would have been permitted to another - is a mamzer. The offspring, however, of another forbidden union - one in which the woman could not have been permitted to another - follows the woman in status. The practical upshot of this mishna is that the child of a gentile woman or a slave girl (neither of whom can contract kiddushin with anybody) is either a gentile or a slave.
If there is a brother and a sister (Reuben and Hanna in my example) and she is a slave, they now belong to two of the different types of relationships delineated from one another in Ketubot. They are siblings, and the penalty for sexual intercourse is karet (Mishna, Keritot 1:1; cf: Leviticus 18:9, 29). That means, based on both Ketubot and Yevamot, their offspring are mamzeirim. And yet, being a slave, she cannot contract kiddushin with any man, which means that their offspring (based on Ketubot) are slaves.