On page 107 of his Algebraic Curves and Riemann surfaces Miranda writes
The definition of a meromorphic or holomorphic $1$-form $\omega$ suggests that in order to define $\omega$ on a Riemann surface $X$, one must give local expression for $\omega$ (of the form $f(z)dz$) in each chart of an atlas for $X$. In fact, one can define $\omega$ by giving a single formula in a single chart. […] A [second] problem may arise, namely that the local expression does not transform uniquely to the other points of $X$. For example, consider the meromorphic $1$-form $\sqrt{z}dz$ defined on the complex plane with the negative axis removed, where the branch of the square root is chosen so that $\sqrt{1}=1$. This can be extended to the negative real axis but not uniquely. Hence we do not obtain a meromorphic $1$-form on the whole of $\mathbb{C}^*$.
I am not sure that I understand this paragraph. Miranda seems to consider the chart $\phi\colon \mathbb{C}\setminus \mathbb{R}_{\leq 0}\rightarrow \mathbb{C}\setminus \mathbb{R}_{\leq0};z\mapsto z$. Denote the corresponding local coordinate by $z$. Then we considers the $1$-form $\sqrt{z}dz$, where $\sqrt{-}$ denotes the principal branch of the square root. Now, what is meant by the above sentence marked in bold? The principal branch of the square root function cannot be extended to a global holomorphic function. Does Miranda refer to the fact that it can locally be analytically continued so that one obtains a holomorphic function $f\colon U\rightarrow V$ such that $\mathbb{R}_{\leq 0}\subset U$ and $f(z)=\sqrt{z}$ for $z\in U\cap(\mathbb{C}\setminus \mathbb{R}_{\leq0})$? And this extension is not unique?
I'm not completely sure what is meant by Miranda. If the meromorphic $1$-form $\omega=\sqrt zdz$ has domain $\mathbb C\setminus\mathbb R_{\le0}$, then it's not possible to extend the domain to any point of $\mathbb R_{\le0}$: suppose $\omega$ can be extended, so that there's an open subset $U$ with $U\cap\mathbb R_{\le0}\neq\emptyset$, a chart $\phi:U\to V$ with $w=\phi(z)$ and $\eta=g(w)dw$ a meromorphic $1$-form on $V$ which transforms to $\omega|_{\mathbb C\setminus\mathbb R_{\le0}}$ under $\phi$. This means that $g(\phi(z))\phi'(z)$ should be a meromorphic function on $U$ and $g(\phi(z))\phi'(z)=\sqrt z$ on $U\cap(\mathbb C\setminus\mathbb R_{\le0})$. This is impossible because if we take a point $\bar z\in\mathbb R_{<0}$, if $z\to\bar z$ from below then $\sqrt z\to-\sqrt{|\bar z|}$ while from above $\sqrt z\to+\sqrt{|\bar z|}$, so $g(\phi(z))\phi'(z)$ cannot be defined in a continuous way on any point of $U\cap\mathbb R_{<0}$ and so is not a meromorphic function on $U$.
On the other hand, if we start with a smaller domain for $\omega$, one which is disjoint from an open subset containing $\mathbb R_{\le0}$, then it's possibile to extend the domain of $\omega$ to also include $\mathbb R_{<0}$ (note that we have no hope for $0$, assuming you want the domain to be connected), but not in a unique way (we can extend the domain "from below" or "from above").