Is "birth tourism," the practice of traveling to another country to give birth and thereby gain citizenship of that country for the baby, real? Indeed it is, though how pervasive the practice is is largely unknown. Many of the families who engage in this practice are Chinese and are seeking both superior U.S. medical care and the advantages a U.S. passport will afford their children. The practice is not illegal unless the mother lies to an immigration officer or overstays her visa, though pregnant patients can be denied entry to the U.S. if an immigration officer determines that they do not have adequate medical insurance. The following articles give some background on the practice:
Marketplace.org's series of articles on birth tourism:
'Birth Tourism,' Fair Path to Citizenship or Legal Loophole?
Arrests have been made for visa fraud related to birth citizenship: