  In this paper, we study the twisted Fourier-Mukai partners of abelian
surfaces. Following the work of Huybrechts [doi:10.4171/CMH/465], we introduce
the twisted derived equivalence between abelian surfaces. We show that there is
a twisted derived Torelli theorem for abelian surfaces over algebraically
closed fields with characteristic $\neq 2,3$. Over complex numbers, the derived
isogenies correspond to rational Hodge isometries between the second cohomology
groups, which is in analogy to the work of Huybrechts and Fu-Vial on K3
surfaces. Their proof relies on the global Torelli theorem over $\mathbb{C}$,
which is missing in positive characteristics. To overcome this issue, we
firstly extend a trick given by Shioda on integral Hodge structures, to
rational Hodge structures, $\ell$-adic Tate modules and $F$-crystals. Then we
make use of Tate's isogeny theorem to give a characterization of the twisted
derived equivalences between abelian surfaces via isogenies. As a consequence,
we show the two abelian surfaces are principally isogenous if and only if they
are derived isogenous.
