It seems you are aware of the basic rule in English that a singular subject requires a singular verb and a plural subject requires a plural verb (cf. "Subject-verb concord", pp 755ff in Quirk, et al. 1985, A Comprehensive Grammar of the English Language (commonly referred to as CGEL), which this answer cites for purposes of reference.
You have set up a scenario where it is impossible (my word) to follow this basic rule. This is because, per CGEL 10.41, "Coordination with or and nor":
The rules are different for subject phrases or clauses which are coordinated with (either...) or...Grammatical concord is clear when each member in coordination has the same number: when they are both singular...the verb is singular; when they are both plural...the verb is plural. A dilemma arises when one member is singular and and the other plural... Notionally, or is disjunctive, so that each member is separately related to the verb rather than the two members being considered one unit, as when the coordinator is additive and. Since the dilemma is not clearly resolvable by the principles of grammatical concord or notional concord, recourse is generally had to the principle of proximity: whichever phrase [is closest] to the verb determines the number of the verb.
You follow this principle of proximity concord with the two sentences you write:
Are the boys or the girl coming?
Is the girl or the boys coming?
However, as I stated in my initial comment, I do not know a native speaker who would say or write the second sentence. Perhaps someone who is a pedant or a grammar nazi would, but I do not hang around such types. This is because in cases such as this, native speakers I know resort to their ear to decide grammaticality. And if the ear says no then they will seek to use a more pleasing-sounding alternative.
Quirk, et al.:
If the [singular/plural] number alternatives for the verb are both felt to be awkward, speakers may avoid making a choice by postposing the second noun phrase or sometimes by substituting a modal auxiliary (cf 10.44).
In your case these tactics result in these changes:
Are the boys coming, or is the girl?
Will the girl or the boys be coming?
Note that the construction will be coming refers to future time as much as is coming.
There are, of course, other ways to rearrange the sentence, but you specifically stated not to. So I stick only with the two strategies mentioned by CGEL to solve this case.
You may be interested this answer to IS or ARE? “The only thing that I want you to hit right now IS/ARE the books”, because it also references proximity concord.