I have troubles with the theory of existence of quasi-conformal homeomorphisms realizing Beltrami coefficients. Let $X$ be a (compact) Riemann surface and $f \colon X \rightarrow \mathbb{C}$ be smooth. Then on a coordinate chart $(U,z)$ of $X$, $x \in X$, the quotient $\left( \frac{\overline{\partial} f}{\partial f} \right)_x$ is of the form $\mu_{f,U}(x)\frac{d\overline{z}_x}{dz_x}$, where $\mu_{f,U}$ is the Beltrami coefficient of $f$ w.r.t. the chart $(U,z)$. This Beltrami coefficient is indeed local and depends on the chart as follows: if $(V,y)$ is another chart with $x \in V$, then $$\mu_{f,U}(x) = \mu_{f,V}(x) \frac{\left( \frac{\partial}{\partial \overline{z}} \overline{y} \right)(x)}{\left( \frac{\partial}{\partial z} y \right)(x)}.$$ In particular, $\mu_{f,U}(x) \ne \mu_{f,V}(x)$ for two different charts. However, the absolute value $|\mu_f|$ is independent of a chart and defines a global function $X \rightarrow \mathbb{R}$.

Now suppose we are given a Beltrami differential $\mu \in L^{\infty}$. Several sources (for instance, "An Introduction to Teichm{\"u}ller Spaces" by Imayoshi and Taniguchi, page 147) claim that there exists a unique quasi-conformal homeomorphism $f \colon X \rightarrow \mathbb{C}$ with Beltrami coefficient $\mu$ (which is supposed to follow from the usual measurable Riemann mapping theorem (mRmt) for $X=\mathbb{C}$). How do I need to understand this statement, because in that textbook they don't actually specify what they mean by a Beltrami differential? Do they mean (a) that $\mu \in L^{\infty}(X,[0,1))$ and that $|\mu_f|\equiv \mu$ or do they mean (b) that $\mu_{f,U}(x)=\mu(x)$ for $x \in U$?

The latter seems wrong to me because we established $\mu_{f,U}(x) \ne \mu_{f,V}(x)$. If it's the first, then is uniqueness really included? (Since we only consider the absolute value...) In this case, I also have difficulties constructing the map $f$ with coefficient $\mu$ with the mRmt. For me, it fails when trying to glue local solutions, which come from the usual mRmt on charts, together.

Cheers