To be born under a veil or behind the veil is a popular term used to describe babies born inside a partial amniotic sac that covers the face, as explained by Baby Med. Another term commonly used for the amniotic sac is a caul, which means helmet.
Babies born under the veil were often believed to have preternatural abilities, such as clairvoyance, explains Baby Med. The caul is almost always harmless to the infant, although the website explains that it must be removed carefully after birth. If the membrane is not removed very carefully from the infant's skin, it may result in permanent scarring.
An article in the Daily Mail tells the story of an infant who was born completely within the amniotic sac, representing an extreme example of the phenomenon. This type of birth is known as a caul birth, while births in which the infant is born with only part of the amniotic membrane covering her face or upper body are called en-caul births.
Baby Med explains that the superstitions surrounding births behind the veil once resulted in parents selling the leftover membrane as a novelty, while some modern hospitals refuse to even notify parents when a child is born this way.