the output won't really be unbalanced, you'll just see some extra 2nd harmonic content if it doesn't cancel perfectly. If the grid signal is not symmetrical, and the output does not have 2nd harmonic content, that kind of implies the transformer is not perfectly symmetrical and it wants that uneven grid drive to make it all come out right.
'course we're assuming the grid signal is not in fact even. A scope to ground on each output tube grid will answer that question. If the input is even in the unit but the sim says its not, the sim is wrong.