How can you tell which noun a clause modifies?

I'm a ESOL teacher, and I'm having trouble answering a question that a student asked me recently. We were going over long sentences, and found this one from the New York Times:

Saudi Arabia said Tuesday that it was halting a nearly month-old bombing campaign against a rebel group in neighboring Yemen that has touched off a devastating humanitarian crisis and threatened to ignite a broader regional conflict.

It's clear to me, as a native speaker, that the clause "that has touched off a devastating humanitarian crisis and threatened to ignite a broader regional conflict" applies to the noun phrase "a nearly month-old bombing campaign" and not the noun phrase "a rebel group in neighboring Yemen," but I can't exactly explain why. I think I know this mostly from the context, but it wouldn't necessarily be obvious to someone learning English.

It would be possible to write this sentence with a clause modifying " a rebel group in neighboring Yemen." Something like, "Saudi Arabia said Tuesday that it was halting a nearly month-old bombing campaign against a rebel group in neighboring Yemen, which has been opposing Saudi influence for nearly three years."

Is it just the comma that makes it clear that one clause applies to the rebel groups, and the lack of comma that shows the clause applies to the bombing campaign?

Any help would be much appreciated! Thank you,

Lee