Matrix Gantt cross over Visual

Hi

I'm looking for a visual that is a combination between a gantt chart and a matrix.  I have a bunch of categories that I want to plot as columns, and then a bunch of systems that fit 1 or more categories that I want to plot as rows.

The matrix visualisation gets most of the way there, but I'm interested in a more visually engaging experience.  Having a bar represent a system and crossing the boundaries of categories is what I'm after.

See the diagram below.  It may just be that there isn't anything like this.  But keen to hear of any suggestions to get as close as possible?

Two areas I see as a challenge are:

* Non-contiguous category membership (as shown by System 7 in the diagram below)

* Most efficient data layout with multiple systems per row where the data allows (as shown by System 2,3 and 4 in the diagram below)

Any help greatly appreciated!