Playing a little bit with Groessencharacters a stumbled on the following question:

Let $K$ be a **non CM** number field with $r_1$ real embeddings and $2r_2>0$ complex embeddings. Set $r=r_1+r_2-1$. Let $\epsilon_1,\ldots,\epsilon_{r}$ be a system of $\mathbf{Z}$-linearly independent (in the multiplicative notation) units of $\mathcal{O}_K$. For $x\in K$ and $j\in\{1,2,\ldots,r_2\}$ let $x^{(j)}$ be the image of $x$ under the $j$-th complex embedding (where the $r_2$ choices of complex embeddings are made so that no complex embedding are related by complex conjugation). 

**Q**: Is it possible to choose $K$ and $\{\epsilon_1,\ldots,\epsilon_{r}\}$ so that the above assumptions are satisfied and such that there exists an $(r+r_2)$-tuple $((m_1,\ldots,m_r);(n_1,\ldots,n_{r_2}))\in\mathbf{Z}^{r+r_2}\backslash\{0\}$ such that
$$
2\pi m_j-\sum_{k=1}^{r_2} n_k\cdot\arg\left(\frac{\epsilon_j^{(k)}}{|\epsilon_j^{(k)}|}\right)=0,
$$
for all $j\in\{1,\ldots,r\}$ ?

This looks like a delicate question about the $\mathbf{Q}$-linear independence of the real numbers $\arg\left(\frac{\epsilon_j^{(k)}}{|\epsilon_j^{(k)}|}\right)$.

**added** The non-CM assumption is essential since for a CM unit $\epsilon_j$, we have $\arg\left(\frac{\epsilon_j^{(k)}}{|\epsilon_j^{(k)}|}\right)\in\pi\mathbf{Q}$. Also, if $\epsilon_j\in\mathcal{O}_K^{\times}$ is chosen to be totally real then $\arg\left(\frac{\epsilon_j^{(k)}}{|\epsilon_j^{(k)}|}\right)\in 2\pi\mathbf{Z}$. So may be one could come up with an example that would combine CM units and totally real units...