Tempest deals with extracting signals at a distance, from things that are already directly broadcasting them in some form or another.

There's no such thing in this case. There is no USB port or VGA cable that is directly carrying the internal state of one of dozens of different processes/applications being run on the target PC.

This work is quite completely different. Using special knowledge of how specific software has been written and compiled, and also with the ability to throw carefully crafted inputs at the software in question, they're able to use some fun statistical methods to guess what the internal state must be.

The paper and this type of work generally is very interesting because software developers are used to thinking of CPUs as "black boxes" which do their thing invisibly and effortlessly.

This kind of work is just as awesome as all the papers on CPU side-channel attacks (cache, branch prediction, etc) which really seemed to take off around 2005.