If they are not subject to our laws then how can we arrest them and deport them?
2 posts above the one to which I replied has the answer (#42).
You also got a sound answer right below your post to which I am replying now.
That is not the meaning of the phrase (subject to the jurisdiction...)
The following link goes to an article that contains quotes from the men who authored the phrase and the debates on the 14th Amendment.
Given the clear language used, by the men who wrote the 14th Amendment and the men who made it law, it seems (to me) that the Supreme Court would be very hard pressed to rule against ending the anchor baby fallacy.