Yes, but what you'd want to do is look for coincidences between a detector for a cosmic ray shower around (above?) the electronics you're monitoring with whatev

Yes, but what you'd want to do is look for coincidences between a detector for a cosmic ray shower around (above?) the electronics you're monitoring with whatever it is these days that instruments ECC events. The time resolution would be pathetic for a nuclear physics experiment, but probably good enough. If you look at the ambient gamma-ray spectrum in a semiconductor detector (which would be germanium rather than silicon) the main background you see is typically from concrete; I'm ashamed to say I've forgotten the energy from K-40, but in the region of 1500 keV. (Ironically, large concrete blocks used for shielding would be regarded as a significant radiation hazard if all the activity in them was concentrated.)