Or, put in examples: which of the following is grammatically correct?
Either you or your sister is going to have to do the chores.
Either you or your sister are going to have to do the chores.
The second option comes off as really strange; am I right to say that the first is grammatically correct because "is" directly follows "your sister" as opposed to "you"?
You are correct; the correct way to write it is based on the number of the closer subject:
Either you or your sister is going to have to do the chores.
As suggested in point 5 on this page, you might consider the order of the subjects to avoid awkward phrases if one is singular and the other is plural. Compare
Neither my father nor my brothers are going to sell the house.
with
Neither my brothers nor my father is going to sell the house.
In short, it is best to recast the sentence.
According to the 3rd edition of Fowler's Modern English Usage (R.W. Burchfield ed.), the verb should follow the number of the first element of a double subject if its elements are coordinated by or. Since the first element is "you" here, it should be "are"; but Burchfield seemed to be thinking rather about whether to use plural or singular in cases like "neither my opinion nor yours matters in this company", where both elements of the subject are singular 3rd person—not so much about cases where each element would require a different form.
About double subjects coordinated by "and", he says that they should normally take a plural verb, unless they are so strongly connected that they may be regarded as one in thought, in which case singular is also possible. He mentions "a certain cynicism and resignation comes along with the poverty of Italian comedy" as a good example.
It seems Burchfield, otherwise a fine chap, is skirting the issue. Credit must be given where it is due: the real Fowler did find the courage. In the 1st edition, he advises the following options, "in order of merit" from top to bottom:
A. Recast the sentence by substituting a modal verb:
Either you or your sister will have to do the chores.
B. Recast the sentence by ellipsis and rearrangement:
Either you are going to have to do the chores or your sister.
C. Have the verb agree with the element nearest to it:
Either you or your sister is going to have to do the chores.
After all these years, Fowler never ceases to amaze with his sense of style combined with common sense. I wish I could write like him—such clarity and vigour. If anyone knows of a modern incarnate, do let me know; he is not always up to date at present (...), and Burchfield does not have his great style.
