No violation of Kirchoff's laws. With the MWBC and a open neutral the thing that changes is applied voltage. Loads that were 120 each with a good neutral are now in series and 240 volts is applied to the series. The load with the higher impedance will have more voltage across it. Input voltage will be 240 and sum of all voltage drops in the circuit will equal applied voltage.
If the neutral is not completely open things are more complex to calculate, as the resistance of the bad neutral likely changes according to conditions, but you are still going to have more drop across the side of the circuit that has the higher impedance.
Exactly. And when you decrease the impedance of the tester, it will see a lower voltage. The equivalent circuit is a 240 volt source with a series impedance which is the impedance of the load on the other side of the circuit. Which we have to assume is fixed for the duration of the test. This is exactly the opposite of what the tester is reporting. Either there is a bug in its program or something very strange is happening in the circuit.
Since the tester (and any real load for a more definitive test) is connected only to two terminals, the entire rest of the system, all the way back through the distribution system if you wish, can be modelled by an equivalent circuit which is either a single voltage source with a series resistor or a single current source with a parallel resistor. Either way, there is no way that drawing more current from those two terminals can cause the voltage at the same two terminals to rise unless the series resistance of the equivalent circuit is a negative resistance. Other than devices like fluorescent tubes, zener diodes and under some conditions motors, you will not find negative resistance in purely passive circuits.
There are some rather esoteric situations which could lead to a condition like the OP's meter claimed it saw, but they depend on non-linear loads somewhere on the other side of the receptacle and are for the most part not consistent with a perfectly normal nominal voltage when no load is applied.
Here is one carefully constructed example of how a current pulse drawn from a DC circuit could cause the terminal voltage to rise:
Imagine a 24 volt battery, with an internal resistance of 1 ohm.
Connected to the + and - of the battery is a voltage divider of two 1200 ohm resistors in series. The load terminals of our network are the top and bottom of the top `1200 ohm resistor. With no load applied, the voltage seen is 12 volts. If you start to draw current, the voltage will decrease. Eventually with a near zero ohm load applied, the voltage will drop to zero and the full 24 volts will appear across the bottom resistor. So far, so good.
This behavior can be modelled by a 12 volt battery with a 600 ohm series resistor.
Now put in parallel with the bottom resistor a circuit composed of an an SCR, zener and resistor such that the SCR will fire and latch when the applied voltage excedes 18 volts. Initially the SCR is not conducting, so the model matches the earlier case exactly.
Look what happens now as you start to draw increasing current from the load points: the voltage decreases toward 6 volts. At that point the voltage on the other resistor reaches 18 volts and the SCR fires. Suddenly the voltage across the lower resistor drops to near zero and the voltage at our load rises to nearly 24 volts.
From this point on, the model of the two terminal network becomes a 24 volt battery in series with a ~1 ohm resistor. Very different.
If all you did was apply a large enough current pulse to the terminals, it would appear to increase the terminal voltage, but that is because of the presence of non-linear active elements which change the equivalent circuit.
I do not think there is any credible equivalent condition on an AC circuit, MWBC or otherwise, which will cause a similar condition.
My current best guess leans toward the presence of a reactive load somewhere in the house which is causing the meter to see a ringing voltage waveform when a step current pulse is applied. Or a programming error in the tester.