This surface is one of several triply periodic minimal surfaces of genus 5 that have vertical symmetry planes over a square grid and diagonal horizontal lines.
It exists as a 1-parameter family, limiting in noded planes and in doubly periodic Karcher-Scherk surfaces.
The divisor of the square of the Gauss map is given below. The red crosses indicate horizontal normal symmetry lines. Given all symmetries, the period problem is 1-dimensional and can be solved using an elementary extremal length argument.
It is amusing to note that the configuration of catenoidal necks in the nodal limit is that of the Horgan surface, which does not exist as a finite total curvature surface.