Because of the nature of the format you have to really be able to anticipate the music changes with ATS in order to keep it matching and catch any accents you want to emphasize. Fat chance and Wild Card (see below) are both usually great at this. here's a performance from the original troupe, Fat Chance BellyDance.
You might have noticed that there is a little R next to ATS on their site. The name is specific to the Fat Chance format, and many tribes have taken the general format and use their own cues, formations, movement combos, and costuming that are unique to their format. For example, Unmata is a popular format, especially with those who lean further into the Tribal Fusion style (come back in two posts for that story ^_~). Wild Card is one of my favorite tribal groups, and they have their own format as well. Any group using the cued improvisation format- with the rotating leader, usually wearing flamenco style skirts, and just about always with finger cymbals- falls under the broader category of Improvisational Tribal style, or ITS, but only Fat Chance's format is ATS. Basically, all ATS is ITS, but not all ITS is ATS. ^_~
This is one of the first youtubes I ever saw of bellydance and is a big part of why I fell in love with it. They are dancing to the introduction to Alf Leila w Leila, a classical Om Kalthoom piece (and my favorite version too!). I fell in love with the women dancing together, the music, and the feeling of a desert caravan, it all grew from there ^_~
You can find more videos on Shem's playlist.