Adermatoglyphia

As reported by the BBC:

The rare condition likely afflicting the Sarker family is called adermatoglyphia. It first became widely known in 2007 when Peter Itin, a Swiss dermatologist, was contacted by a woman in the country in her late twenties who was having trouble entering the US. Her face matched the photograph on her passport, but customs officers were not able to record any fingerprints. Because she didn't have any.

Upon examination, Professor Itin found the woman and eight members of her family had the same strange condition - flat finger pads and a reduced number of sweat glands in the hands. Working with another dermatologist, Eli Sprecher, and graduate student Janna Nousbeck, Professor Itin looked at the DNA of 16 members of the family - seven with fingerprints and nine without.

"Isolated cases are very rare, and no more than a few families are documented," Prof Itin told the BBC... Prof Itin dubbed it "immigration delay disease", after his first patient's trouble getting into the US, and the name stuck...

Amal and Apu recently got a new kind of national ID card being issued by the Bangladeshi government, after presenting a medical certificate. The card uses other biometric data too - retina scan and facial recognition.

But they still can't buy a Sim card or obtain a driver's licence, and obtaining a passport is a long and drawn out process.

In discussions of this syndrome, comments are sometimes made that the afflicted family members would be excellent thieves.  But the counterpart of that is that they would be arrested on suspicion for every crime in which no fingerprints were found.