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The error in simulation setup: It apparently uses MOSFETs without backward (substrate) diodes. They don't exist as technical devices.
In a real inverter, the diodes clamp the inductor flyback voltage to the power supply.
Yes. I neglected to install these.
After revising my simulation, the spikes are now reduced (as you state).
At switch-Off, the coil kicks current back up through the power supply. Hence the power supply needs to contain a component in its output stage, which can offer low impedance to this reverse current.