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For certain real quadratic fields $K$ with sufficiently small discriminant weproduce explicit unit generators for specific ray class fields of $K$ using anumerical method that arose in the study of complete sets of equiangular linesin $\mathbb{C}^d$ (known in quantum information as symmetric informationallycomplete measurements or SICs). The construction in low dimensions suggests ageneral recipe for producing unit generators in infinite towers of ray classfields above arbitrary real quadratic $K$, and we summarise this in aconjecture. There are indications [19,20] that the logarithms of thesecanonical units are related to the values of $L$-functions associated to theextensions, following the programme laid out in the Stark Conjectures.

Generating Ray Class Fields of Real Quadratic Fields via Complex Equiangular Lines