I went looking for examples of what you were talking about and discovered that, amazingly enough, a single human voice can even reliably produce overtones! I'm guessing these are very different, though, than what you're talking about https://www.youtube.com/watch?v=2i61_JNc_Nc&t=1m25s
I'd love to hear some examples of a barbershop n-tet doing it too, I'm sure it's even better. But when I search for 'barbershop overtones' it's all music by groups called The Overtones :)
The principle is actually the same -- she is modifying her singing apparatus to emphasize different upper partials that are already in her voice. Barbershoppers do this as well, though less explicitly (we do vowel matching, which helps emphasize upper partials to produce greater ring).
Here's a video from a perennial favorite quartet (the Gas House Gang): https://www.youtube.com/watch?v=pvYT_yWiLqU The top/tenor note is often the same note as the primary overtone produced by the other 3 parts (adding further emphasis to the overtone), but this effect is what gives barbershop the quality of sounding like more than 4 voices, and produces the "ring" in the sound.
Just had to add (since I'm a giant Barbershop Geek in addition to all of the other ways I'm a geek ;))...
Lest anyone think that barbershop is just for old folks:
https://www.youtube.com/watch?v=XWmFfFx24zs (Vocal Spectrum, 2004 international collegiate quartet champions and 2006 international quartet champions)
I can do it pretty well -- the higher one, not the deep version (I can approximate the growly version, but it'll have me coughing after about 10 seconds, so I'm clearly doing something wrong!) -- after seeing Huun Huur Tu in concert back when I was in school I just tinkered until I could get the overtones reliably.
I also did a bit of barbershop (we called ourselves The Overhead Projectors...) and playing with throat-singing was actually quite useful in that context. I'm not sure how much it affect my actual voice control (probably some), but mostly I just became keenly aware of how much tone could vary just based on little changes in positioning of mouth, throat, tongue, palate, etc..
Overtone singing itself isn't very common in barbershop music, but their harmonies certainly make use of overtones. Here's a performance of a song called Play That Barbershop Chord (a stable in the genre). It's not instructive, but it's got some great harmonies: https://www.youtube.com/watch?v=F5bYnRkwpco
Here's a fairly technical article about three different seventh chords, one (the harmonic seventh) being the so-called barbershop seventh chord. There are embedded YouTube videos with computer-generated audio samples. http://www.garygarrett.me/?p=1575
There are some multi-track barbershop videos on Youtube which are good examples of over-the-top overtones. The fact that it's the same person (with the same timbre, "overtone profile") doing all the voices seems to amplify the effect. Here's one: http://www.youtube.com/watch?v=QpU_X6VKqvA
Here's a great video of the Ambassadors of Harmony (an incredibly good barbershop chorus from St. Louis, MO) doing some exercises to tune overtones, and they're really loud: https://www.youtube.com/watch?v=sCdQVqQXkzc
Listen for when the director sings the overtone note alone, then listen to the chorus as they produce it.
Incidentally, cheesy musical puns are the norm for barbershop and other a capella groups. Aural Fixation, Decibelles, Chiefs of Staff, Vocal Majority, Clef Hangers, and so on.
I'd love to hear some examples of a barbershop n-tet doing it too, I'm sure it's even better. But when I search for 'barbershop overtones' it's all music by groups called The Overtones :)