Most people will have see a geometric "explanation" of the addition law on elliptic curves given by embedding it as a cubic in the projective plane and cutting it with lines.

Is there a similar explicit, geometric definition of the addition law on (a family of?) abelian surfaces?

So the question is really: Give a nice embedding of abelian surfaces into projective space and then define the addition law using this embedding - if not for all abelian surfaces, at least for some non trivial family.