In his 1992 paper about doubly periodic minimal surfaces, Fusheng Wei also proves the existence of a 2-parameter family of triply periodic surfaces of genus 4 of which his doubly periodic surfaces are limits.
Below are two views of the same surface, only assembled differently to show the inside resp. the outside.
Besides the genus 2 limit, there are other limits as well, for instance vertical planes over a rhombic tiling, with the intersections resolved by singly periodic Scherk surfaces of different periodicity:
Below is an animation showing the deformation family for surfaces in a box with a square base:
F.Wei: Some existence and uniqueness theorems for doubly
periodic minimal surfaces, Invent. Math 109 (1992), 113–136.