  In \cite{MP} we have shown that if a compact Riemann surface admits a Strebel
differential with rational periods, then the Riemann surface is the complex
model of an algebraic curve defined over the field of algebraic numbers. We
will show in this article that even if all geometric data are defined over
$\overline{\mathbb{Q}}$, the Strebel differential can still have a
transcendental period. We construct a Strebel differential $q$ on an arbitrary
complete nonsingular algebraic curve $C$ defined over $\overline{\mathbb{Q}}$
such that (i) all poles of $q$ are $\overline{\mathbb{Q}}$-rational points of
$C$; (ii) the residue of $\sqrt{q}$ at each pole is a positive integer; and
(iii) $q$ has a transcendental period.
