The problem with that is that unlike Thai food and Indian food, there is no single dish that you can basically say epitomizes Filipino food in the minds of many -- it's quite varied, and doesn't rely on a core spice combination or anything like that.
The closest I've seen to a "signature dish" is probably adobo, which is one of the simplest dishes to make in the Filipino repertoire, and therefore, one of the most widespread (I've seen this at Hawaiian restaurants like L&L in addition to Filipino chains).
This being said, Wikipedia is generally a good place to start when searching for resources and references on topics like national cuisine (use the Cuisine of the Philippines article heading - I don't have enough posts to include URLs yet). That particular article is reasonably comprehensive.
As for Filipino franchises abroad, they're sort of thin on the ground, but in Southern California, I can think of several:
Goldilocks Bakeshop (16 US locations)
Jollibee (11 in California, 1 in Nevada)
Manila Good-Ha (several in California)
Red Ribbon Bakeshop (15 in California)
And individual places:
Barrio Fiesta
Manila Sunset
All these are findable on Google.
Outside of Southern California (and one restaurant, Ma-I Island Grill, conveniently located within walking distance of Arizona State University), I'm afraid I'm really of no help with finding Filipino places elsewhere.
Wow, kinda off-topic for a Japanese forum. Oops.