A refinement of the rank 1 Abelian Stark conjecture has been formulated by B. Gross. This conjecture predicts some ${\frak p}$-adic analytic nature of a modification of the Stark unit. The conjecture makes perfect sense even when ${\frak p}$ is an Archimedean place. Here we consider the conjecture when ${\frak p}$ is a real place, and interpret it in terms of 2-adic properties of special values of L-functions. We prove the conjecture for CM extensions; here the original Stark conjecture is uninteresting, but the refined conjecture is nontrivial. In more generality, we show that, under mild hypotheses, if the subgroup of the Galois group generated by complex conjugations has less than full rank, then the refined conjecture implies that the Stark unit should be a square. This phenomenon has been discovered by Dummit and Hayes in a particular type of situation. We show that it should hold in much greater generality.