The classification of real quadratic fields which satisfy Hammarhjelm's condition

A real quadratic field satisfies Hammarhjelm's condition if its ring of integers has unique factorization, and the Minkowski lattice of its ring of integers contains no point in a certain rectangle determined by the fundamental unit. Such fields have recently appeared in the study of visible points in algebraic cut-and-project sets. We prove that there are exactly seven real quadratic fields satisfying Hammarhjelm's condition, namely those with discriminant 8, 5, 13, 29, 53, 173, 293.
The proof is based on showing that for such fields, the fundamental unit is small relative to the discriminant, together with genus theory and Biro's classification of class number one fields in Yokoi's family.