This is a special case of this question, where it is asked whether there exists an algorithm to determine whether two varieties are isomorphic. There, an answer by Bjorn Poonen explains how to solve the problem for curves, and says that such an algorithm is known when the varieties involved are of general type, but says that the existence of an algorithm to handle all cases is open.

I am interested in whether such an algorithm is known for the case when the varieties involved are abelian surfaces, in particular when the surface is not given as the Jacobian of a genus 2 hyperelliptic curve (consider for instance a product of two elliptic curves, or when a projective embedding is given). I do not know if the problem is difficult, but it is not obvious to me how to devise such an algorithm, and I would appreciate if someone could point me in the right direction.

It may be worth pointing out that it suffices to decide if the two surfaces are birational, so an algorithm to decide whether two surfaces are birational would already solve the problem. However this question is also unanswered.