In December, Hector Villalobos traveled from Colorado to his native Mexico for an interview, part of his application for U.S. permanent residency. Mr. Villalobos expected to be gone a couple of months to complete the process.

Seven months later, U.S. consular officers haven't allowed the 37-year-old handyman to return home to his wife and three children. The problem: tattoos—some associated with violent Mexican gangs—on Mr. Villalobos's body.

"He...