One can add a handle to the Schwarz CLP surfaces and keep the horiontal straight lines. This gives a 1-parameter family of triply periodic surfaces of genus 4.
The images above show four copies of a translational fundamental piece each. One limit consists of two singly periodic Scherk surfaces with normal speed and double speed (i.e. the translational period is halved), respectively. The other limit consists of a double periodic Scherk surface and a doubly periodic Karcher-Scherk surface of genus 1.
The divisor of the squared Gauss map on the quotient torus is shown below. The parameters a and τ need to be adjusted to solve a 1-dimensional period problem which can be visualized in terms of the translation structures of G dh and 1/G dh by requiring that they fit together as shown below (corners correspond to the points of the divisor).
There is numerical evidence that these surfaces can be deformed into Wei’s triply periodic surface of genus 4.
D. Freese, M. Weber, A.T. Yerger, R. Yol: Two New Embedded Triply Periodic Minimal Surfaces of Genus 4