Driving LM386 with signal referenced to virtual ground
Connect your "virtual ground" (or as I would call it an AC ground) via a capacitor to real ground. This prevents the DC levels on your virtual ground getting upset by a conducting connection. Make sure the capacitor value is reasonably high; in your circuit you are using 1 kohm input resistors so, at 20 Hz, the capacitors impedance should not be higher than 1 kohm. Sounds like 10 uF should do the job. I'd use a non-polarized one although it will probably be OK with an elecrolytic.

You may need to put a 10 uF capacitor in series with each 1 kohm resistor too.