I need to find solutions to the Beltrami equation

$$ \frac{\partial w}{\partial\overline{{z}}}=e^{i\varphi(z)}\frac{\partial w}{\partial z} $$

for $w=w(z,\overline{z})$ and $\varphi(z)$ some given, real, harmonic function. So the Beltrami coefficient is just a phase.

Among the almost infinite literature about the Beltrami equation the only thing I've been able to find is that the solution exists and is unique, provided that $w$ is specified along a compact contour on the plane. Solving it numerically is possible, but I'd like to have a deeper understanding of the solutions, and in particular, what are the implications of $\varphi(z)$ being harmonic, and even more, of the Beltrami coefficient being a phase, $|\mu(z)|=1$, a fact that certainty drives the problem.

Any ideas ?