Consider meromorphic functions $f_j:\mathbb{C}\setminus[0,\infty)\to\mathbb{P^1(C)}$ for $1\leq j\leq n$, with branch points at $0$, $\infty$, $1$ namely monodromy matrices $M^{(0)}$, $M^{(\infty)}$ such that
$\lim\limits_{\epsilon\to 0}f_j(x+i\epsilon) = \lim\limits_{\epsilon\to 0}\sum_{k=1}^{n} M^{(0)}_{jk} f_k(x-i\epsilon)$ for $0<x<1$,
$\lim\limits_{\epsilon\to 0}f_j(x-i\epsilon) = \lim\limits_{\epsilon\to 0}\sum_{k=1}^{n} M^{(\infty)}_{jk} f_k(x+i\epsilon)$ for $1<x<\infty$.
Clearly, multiplying all $f_j$ by the same meromorphic function $\mathbb{P^1(C)}\to\mathbb{P^1(C)}$ does not change monodromy matrices. Is the converse true? Namely, if $(g_j)_{1\leq j\leq n}$ with branch points at $0$, $1$, $\infty$ have the same monodromy matrices as $(f_j)_{1\leq j\leq n}$, is it true that $f_j=\Lambda g_j$ for all $j$, with $\Lambda:\mathbb{P^1(C)}\to\mathbb{P^1(C)}$?
There may exist a useful rephrasing of this question in terms of Riemann–Hilbert problem perhaps.