I thought, "They must be doing something wrong with their measurement setup," so I gave it a go with a uCurrent Gold. Much to my surprise, I got the same results.
A few notes:
1. This happens even with a differential probe at the output
2. This happens even at DC with an isolated supply.
3. This happens even by simply shorting the negative output terminal to the negative input terminal; connecting the negative output terminal via a separate path to a low-impedance ground is not necessary
4. Careful measurements with a DMM placed on the high side and the low sides of the path back to the signal generator confirms that the true current remains steady on both the high and low sides regardless of whether the two negative terminals on the uC Gold are shorted. (This measurement required a differential probe on the uC Gold output, obviously)
5. This does not happen if the short between the two uC Au negative terminals is switched to even a 1 ohm resistor
6. The lead I was using for a short had a measured resistance of 10 mohm. Frequency was 95 Hz, amplitude was 1 Vrms, load was 10 ohms in parallel with the uC gold.
7. This does not happen on the other uC Gold ranges; amplification is the expected 100:1.
8. This is probably not related to the bias current on the first-stage uC Gold op-amp, since those are CMOS inputs with bias currents in the picoamp range.
9. Shorting the two negative terminals also introduces a small DC offset to the output, which suggests that the issue is somehow related to the virtual ground.
10. With the two negative terminals "shorted" with the 10 mohm-resistance lead, and differential probes on the output, the potential difference between the negative output terminal and the negative input terminal is about -500 nV; i.e., a few microamps of current is flowing out of the negative output terminal towards the negative input terminal. (Or so I think. This is right down near the limit of what I can measure reliably, and I don't have a meter with a small enough sense resistor at the microamp range to measure the current directly.)
I've considered a number of explanations, but none of them have held up to experimental measurement with an actual uCurrent Gold in the system.
I suspect this is somehow related to the trace resistance, but again, nothing I've worked through has stood up to experimental evidence.
No solutions, just some observations.