12AT7 Phase splitter

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Try connecting directly at the cathode. That takes the voltage divider of the 400 ohm / 10k resistor out of the signal path. That voltage divider creates about a 4% drop. It's still not going to be a perfect match between the phases, but it will be closer.

And as mentioned previously, the load resistor (after the decoupling cap) should be at least 10x larger than the load resistor(s) in the DC current path of the tube. 10k at the plate/cathode, 100k after the cap, minimum.
 
Try connecting directly at the cathode. That takes the voltage divider of the 400 ohm / 10k resistor out of the signal path. That voltage divider creates about a 4% drop. It's still not going to be a perfect match between the phases, but it will be closer.
It's the other way around. The output should be taken across the 10k cathode resistor only, so it exactly matches the anode resistor, yielding perfect balance every time (within the tolerance of the resistors). If you take it directly from the cathode the balance will be worse. The OP doesn't have a balance problem, it is a misunderstanding caused by comparing adjacent peaks instead of opposing peaks.
 
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