Minimize the Cost Function of Values of Vectors Based on Their Amplitude
Since there is no prior at the Vector level this is basically element wise problem.
Moreover, if we assume the noise to be White Noise with zero mean then the answer can be very simple.
Since the phase difference is always a multiply of 180 [Deg] we can, without loss of generality, assume they are on the real axis.
So what we have can be modeled as:
$$ {z}_{1} = {w}_{1} + {n}_{1}, \; {z}_{2} = {w}_{2} + {n}_{2}, \; \left| {w}_{1} \right| = \left| {w}_{2} \right| $$
So the answer will be $ \hat{w}_{1} = \operatorname{sign} \left( {z}_{1} \right) \frac{\left| {z}_{1} \right| + \left| {z}_{2} \right|}{2} $ and $ \hat{w}_{2} = \operatorname{sign} \left( {z}_{2} \right) \frac{\left| {z}_{1} \right| + \left| {z}_{2} \right|}{2} $.
This comes from a simple intuition. Let's assume bot are positive and $ {w}_{1} = {w}_{2} $. Then of course the answer will be, for any white noise, to average the results. So we are doing the same, averaging them, just taking care of the case they have opposite phases (In the complex plane they are on the same circle just could be on opposite directions).
If you assume AWGN and build the Maximum Likelihood Estimator you'll get the same result.