It is possible to add a handle to Batista’s triply periodic Costa surface, as shown by Valerio Ramos-Batista and Plinio Simões in 2008.
This surface of genus 7 comes in a 1-parameter family with a rectangular period lattice. One limit is CHM1,3, a singly periodic Callahan-Hoffman-Meeks surface with handles.
The other limit is the doubly periodic Scherk surface, with the handles disappearing into the ends:
Resources
Paper by Simões and Batista
Minimal Surfaces 