In his 2009 PhD thesis, Peter Connor found many numerical examples of doubly periodic minimal surfaces whose existence awaits proof. The most challenging genus 3 example appears on page 84:
The surface comes in a 1-parameter family which requires solving a 5-dimensional period problem, because it has now symmetries besides reflections at the vertical coordinate planes and lattice translations. The conformal parameters are a magnitude of 1010 apart, which raises philosophical concerns about physical existence of such surfaces, given that differences in orders of magnitude of physical quantities appear to be bounded.
More to the point: These surfaces are hard to come by. The player in the game is the top of the two catenoidal handles connecting the vertical layers. Moving it just slightly up pulls its cousin, the bottom catenoidal handle, up as well:
What happens when the handle is pulled up further is not clear. My guess is that the lower catenoidal necks will be pinched, the top portion become a doubly periodic Karcher-Scherk surface, and the bottom portion an ordinary Scherk surface.
Similarly, when the top neck is pushed down, I expect the same result, with top and bottom rôle reversed.