"There is some question as to whether someone who has no recognized identity and isn't even supposed to be here can be 'subject to jurisdiction' of the US."
Subject to the jurisdiction means subject to the laws of the US which every person in the US is except foreign diplomats.
Suppose a baby is born to an illegal mother without record of the birth being made. The mother claims it was born inside our borders.
Is it a citizen or not?
E]very person born within the limits of the United States, and subject to their jurisdiction, is by virtue of natural law and national law a citizen of the United States. This will not, of course, include persons born in the United States who are foreigners, aliens, who belong to the families of ambassadors or foreign ministers accredited to the Government of the United States, but will include every other class of persons. It settles the great question of citizenship and removes all doubt as to what persons are or are not citizens of the United States. This has long been a great desideratum in the jurisprudence and legislation of this country http://judiciary.house.gov/legacy/6042.htm