If the true ground connection at the power supply is the ONLY ground connection it will not be a problem.
The real issue comes with what you do with your signal outputs. Consider the schematic below.
simulate this circuit – Schematic created using CircuitLab
A simplified version of your virtual ground circuitry is shown on the left with the power supply grounded on the negative pin. That part should work as expected.
However, when you feed the output if the first circuit to a second box, or somewhere else in the same box, and that second part also has a ground connection to the real ground you create a short between your virtual -12V and ground.
At that point, at best, the circuitry no longer functions as expected, at worse you can exceed the input voltages on the right hand circuit and burn something.
If your left hand circuitry is stand alone and does not connect to any other device, then power supply isolation should not matter. I say should, because if your power supply is not isolated and your gizmo is in a metal box that is grounded, you need to make sure the virtual ground does not touch that case anywhere.