Let $S$ be the unit circle in the complex plane, $$ S = \{z \in \mathbb{C} : |z| = 1\}. $$
For values $z_1^{(1)},z_1^{(2)},z_2^{(1)},z_2^{(2)},\ldots,z_k^{(1)},z_k^{(2)} \in S$, letting \begin{align*} A &:= \left|\sum_{j=1}^k z_j^{(1)}z_j^{(2)}\right| \\ B_1 &:= \left|\sum_{j=1}^k z_j^{(1)}\right| \\ B_2 &:= \left|\sum_{j=1}^k z_j^{(2)}\right|, \end{align*} do we necessarily have that \begin{equation} \hspace{30mm} A \geq B_1 + B_2 - k \, ? \hspace{30mm} (1) \end{equation} If not, do we necessarily at least have that \begin{equation} \hspace{30mm} A^2 \geq B_1^2 + B_2^2 - k^2 \, ? \hspace{26mm} (2) \end{equation}
Note that (1) does indeed imply (2): the triangle inequality gives that $B_1$ and $B_2$ are each at most $k$, and hence $$ [\text{RHS of (1)}]^2 - \text{RHS of (2)} \ = \ 2(k-B_1)(k-B_2) \ \geq \ 0. $$
Context: For multivariate data where each coordinate is an angle, it is common to measure the coherence (resp. squared coherence) between two of the coordinates across the data set, by taking the magnitude (resp. squared magnitude) of the mean of the complex exponential of the difference between the two coordinates. The question is then essentially whether one minus this value defines a metric on the set of coordinates (with Eq. (1) being for coherence and Eq. (2) being for squared coherence). This is a highly relevant question both for phase-coherence of stationary stochastic processes as defined in https://en.wikipedia.org/wiki/Coherence_(signal_processing), and for various notions of time-localised phase-coherence such as in Appendix B.1 of https://www.sciencedirect.com/science/article/pii/S1063520320300750.
There are counterexamples for $k=2$: let $$z_1^{(1)}=z_1^{(2)}=(1+i)/\sqrt2\ \ \text{and}\ \ z_2^{(1)}=z_2^{(2)}=1,$$ $$\Rightarrow\ \ A=\left|\frac{(1+i)^2}2+1\right|=\sqrt2\ \ \text{and}\ \ B_1=B_2=\left|\left(1+\frac1{\sqrt2}\right)+\frac i{\sqrt2}\right|=\sqrt{2+\sqrt2}.$$ Then $A^2=2<2\sqrt2=B_1^2+B_2^2-k^2$, which means $(2)$ is false and thus $(1)$ is false.