My charger and a pals run fine with engine on or off .... both are versions of the B6 .... But what happens when engine starts is the voltage error kicks in with the fluctuation ... press stop ... then start ... fine.
To hayofstacks ..... an alternator should be pushing out more than 14V otherwise it will have difficulty overcoming battery resistance / EMF etc. At 13.x - you will never have a well charged battery on your car.
I spend loads of time on boats sorting out charging of 12v batterys because we rely on charged batterys for domestics / services / instruments etc. If I had an alternator only pumping out 13.x volts - I'd put it into service to get it fixed.
The only time I see low voltage like that is when a relay or especially a diode controlled suply is involved. A diode will typically drop the voltage by about 0.6 - 0.7V ... a relay if an old one can cause a resistance drop of up to 0.5V ....
My 12v socket in back of my Volvo XC shows 14.5V when engine running ... which is expected with voltage drop along the wiring etc.
My B6 charger is stated to be suitable for up to 18V supply.