In order to do exact probabilistic inference on real life network efficiently, one must find a good Tree Decomposition of the network. This process is known as the Junction Tree algorithm. It's a bit hard to visualize the result, but while browsing tree decomposition graph pages on Wikipedia, I got an idea. Instead of bags with variables, we plot it as a collection of colored strips where each strip corresponds to a variable and Running Intersection property guarantees there will be no breaks.
Here's the result for the width-7 tree decomposition of the moralized Barley network