The surfaces resembles the Schwarz CLP surface with handles added. The divisor of the squared Gauss map is below.
Below are 8 copies of a translational fundamental piece near the two limits: Two doubly periodic Karcher -Scherk surfaces of genus 1 and 2 on the left, and singly periodic Scherk surfaces with different frequency.
Resources
Mathematica Notebook
PoVRay Sources
Minimal Surfaces 