Wow. The springs on the rear inner pads are what hold it to the pistons in the calipers. Seems like someone was lazy while they were installing those.

I had a similar noise recently after replacing my rear struts. My issue is the lower bolts weren't torqued enough. 70 ft/lbs later and they're quiet as can be.

 

My 2006 Mazda 3 has this same problem - a sort of metallic rattle when going over bumps when the brakes are *not* being applied (noise disappears when the brakes are in use). The noise began after I had the brakes replaced a little over a year ago. After taking the car back and getting no help whatsoever, I ended up taking the car to a different mechanic who said the brake pads had been "put on dry." They fixed that - installed a new suspension (which I had requested and the car desperately needed) and fixed the exhaust shield(s).

The noise is still there.

Spring clip issue?

 

If it comes and goes depending if the brakes are applied, I'd definitely suspect the pads. You can install the pads with a dab of high temp silicone to glue them to the caliper, which may help if the springs aren't working right / worn / broken.

On a similar noise note, I just had a rattle start in the rear, only there when hitting bumps, pavement transitions. It sounded somewhat heavy in the car but sounded like a washer on a bolt with the windows down. I tracked it down by hitting the suspension parts by hand and found that the bottom-most 1/3 length of the coil spring broke off and was rattling over bumps.