This is a follow-up question to
A lot of well known codes (5 qubit code, 7 qubit Steane code, 9 qubit Shor code) have logical zero and logical one which can be written as (a global scalar times) a linear combination of computational basis kets with only $ \pm 1 $ as coefficients.
The question linked above shows that any stabilizer code has codewords which can be written as (a global scalar times) a linear combination of computational basis kets where every coefficient is $ \pm 1, \pm i $.
I'm curious about these $ \pm i $ coefficients. Does anyone know any stabilizer codes which seem to use $ \pm i $ in an essential way? In other words
What is an example of a stabilizer code which has codewords with some $ \pm i $ coefficients and is not equivalent by local unitaries to a stabilizer code with just $ \pm 1 $ coefficients?
In general I'm interested in any examples of cool stabilizer codes that use $ \pm i $ relative coefficients.
Note: Corollary 2 of Thm 9 in https://arxiv.org/abs/1711.07848 says some pretty cool stuff (although part (iv) of the corollary is wrong). In particular, part (iii) of Corollary 2 says that the number of $ \pm i $ amplitudes is either 0 or it is half the number of non-zero amplitudes.