Commentary for Kiddushin 116:13

In this case, Reuven makes an ambiguous statement, “Be betrothed to me from now and after thirty days.” It is unclear whether his betrothal begins now, or after thirty days. Alternatively, she may begin to be betrothed now but not fully betrothed until thirty days. In any case, if Shimon comes along and betroths her within the thirty days, his kiddushin are also doubtfully valid. If Reuven’s betrothal has begun, then she is betrothed to Reuven and Shimon’s act is irrelevant; but if Reuven’s betrothal has not begun, then she would be betrothed to Shimon. Alternatively, if Reuven’s betrothal has begun but not been completed, she may be betrothed to both of them at the same time. In such a situation she would be forbidden to both and require a get from both (see Gittin 7:3). If she was the daughter of a priest and one of them was an Israelite, she would no longer eat terumah lest her marriage to that man was valid. Similarly, if she is the daughter of an Israelite and one of the men was a priest she would not eat terumah lest her marriage to that man was not valid. In other words, we act stringently and she doesn’t get to eat terumah no matter what the case. Again, this is the mishnah’s way of saying that she is doubtfully married to both men and not fully married to either.