Is "Ram and Shyam is going to the fair" correct?

Why

Abhay's rule about matching the nearest word can actually work in some more complicated sentences, so it's not a baseless guess — but this particular sentence has a simpler, more direct rule that takes priority. When two singular subjects are joined by "and", they combine into one plural subject together — "Ram and Shyam" means two people doing the action jointly, exactly like saying "they", which everyone agrees takes a plural verb. The "and" is doing real grammatical work here: it's adding the two subjects together into a group of two, not just listing them side by side while the verb quietly ignores one of them. Test: replace "Ram and Shyam" with the pronoun it stands for — "They are going to the fair" — and "they is" immediately sounds wrong, confirming the combined subject needs the plural verb. His sister is right: two subjects joined by "and" form a plural subject together, so it's "Ram and Shyam are going", not "is going".