I don't know if this situation has a particular name. I'll just give the example and my question.
Suppose we have two naturally concurring groups, $A$ and $B$. By that, I mean there is a naturalistic mechanism for why the groups are separated and that the groups $A$ and $B$ are (approximately) independent. I.e., a researcher did not assign them this label. However, suppose that in group $A$, a treatment ($T = 1$) was randomly given to the members of the group with the intent to measure the effect of the treatment. However, the researchers lost their documentation on who in group $A$ received the treatment and who received the control. All they know was that the treatment was assigned completely at random in group $A$ to, say, half of the members. They are also sure that no one in group $B$ received treatment.
Can the researchers still estimate a treatment effect somehow by taking the labels $A$ and $B$ as the treated and controlled group? My thinking is perhaps no, but I was curious anyway.
Edit: I forgot to mention that it is a reasonable assumption that groups $A$ and $B$ come from the same super-population. The only difference is in their labels and that group $A$ received a (random) treatment and that group $B$ did not.