Yevamot, Chapter Fifteen, Mishnah Six
This mishnah discusses a case where a man was married to more than one wife, and one wife claimed that he had died and the other wife was not able to know whether this was true.
1) If a woman and her husband went to a country beyond the sea, and she returned and stated, my husband is dead, she may be married again and she also receives her ketubah.
a) However, her rival wife is forbidden to remarry.
1) If [her rival wife] was the daughter of an Israelite [who was married] to a priest, she is permitted to eat terumah, the words of Rabbi Tarfon.
2) Rabbi Akiva says: this is not a way that would lead her away from transgression, unless [it be enacted that] she shall be forbidden both to marry and to eat terumah.
In mishnah four we learned that a rival wife may not testify that a womans husband is dead. Therefore, in the situation in our mishnah, although one wifes testimony is believed and she is allowed to remarry and collect her ketubah, the rival wife may not remarry. Unlike the previous mishnah, where the rival wife explicitly stated that he was not dead, in our situation, since the husband and the other wife were overseas, the wife who remained behind cannot know whether or not he is still alive.
Rabbi Tarfon and Rabbi Akiva disagree over whether or not the rival wife may eat terumah after the other wife has stated that the husband is dead. This debate is only critical if she was from an Israelite family, she was married to a priest, and she had no children. In such a case, if her husband dies she is not allowed to eat terumah. According to Rabbi Tarfon, the halakhah is consistent: just as she is treated as if she is still married and not allowed to remarry, so too she may eat terumah, as if her husband was still alive.
Rabbi Akiva holds that allowing her to continue to eat terumah might lead her to transgress, because her husband might very well be dead. Rather, Rabbi Akiva is strict on both counts: she may not remarry lest her husband be alive but she may not eat terumah lest her husband is dead.