What is this tea stalk that almost every cup of tea in anime has?
One type of tea style popular in Japan is Karigane (http://en.wikipedia.org/wiki/Karigane_(tea)#Karigane) Karigane tea is made from the stalks, not the leaves. This means anything that slips out of the teapot's strainer will be a stalk. This could be one of many types of Japanese tea: you can find Gyokuro Karigane (the stems and veins of Gyokuro tea leaves) or Hojicha/Houjicha Karigane (stems of tea leaves roasted in the same way as standard Houjicha.)

Stem teas tend to be a lighter, more delicate version of their heavier-hitting leafy counterparts.



You can also find blended teas, where there will be leaves as well as stems in the mix.

P.S. I like the "natural habitat" line :)