Turns out that there are some prerequisites for starbase modules that aren't made clear in the tech tree. According to the manual, in order to build the module Interstellar Consortium, on a starbase you must first build Xeno Concert Hall.

There's no way of determining this in-game. I was merrily researching down the Cultural Domination tech path, and when I sent my constructors out to build a mega-influence starbase, I found I could only build the first two Influence modules (Diplomatic Outpost and Foreign Relations Centre).

After asking around on #galciv and searching the forums I eventually found a reference to the section in the manual that lists modules. In there, I see this Xeno Concert Hall and as a prereq for that, Galactic Mall.

It's just my luck that Galactic Mall comes from the Advanced Trade tech, which isn't available in the campaign mission that I'm playing!

So all that research and those constructors that I've built were actually a great big waste of time.

Please upgrade the tech tree so that it properly describes the prerequisites for starbase modules. I envisage something like:

STARBASE MODULES
Xeno Concert Hall