Here's some food for thought, years ago I ran across a Leeson FHP series AC inverter that claims to be able to drive single or three phase induction motors. see
http://www.leeson.com/TechnicalInformation/pdf/FHPSeriesChassisMountVSpeedDrives.pdfIf you look in the above document on pages 28 & 29 you should see how to use it with a single phase motor both with and without a phase shift capacitor.
I have one of these inverters but it has an LV power supply problem so I haven't been able to try it out. I did try another brand 1 hp inverter with a capacitor run motor without the capacitor and it does indeed work.
I recall analyzing a two phase motor driven by a 3 phase inverter with the winding common connection direct to one of the phases and, if memory serves, the voltages on the phases are 60 degrees apart relative to the common connection. While not the desired 90 degrees it should be good enough to be useful.
If the common connection's voltage swing is reduced the angle between the other two phases is increased. I believe 87% of the other two will give a net 90 degrees. If one was building their own inverter it should be possible to have the voltage on the common phase reduced by reducing the PWM duty cycle variation but it can be done with transformers. Look up Scott-T connection and imagine using a tap on the non-center tapped transformer to reduce the common phase voltage.
I'll bet there will be a fair ripple voltage at the motor run frequency on the DC link supply due to the unbalanced load currents pumping the DC voltage up. ie., when the two winding currents are negative, these currents have to go back into the inverter and are pushed back into the DC supply capacitor causing its voltage to rise.
As an aside, I used a washing machine motor many years ago on a small drill press. I got two speeds plus forward and reverse. I wonder if the motor was wound for a Dahlander or consequent pole connection. ie., switchable number of poles. I gave it away so I'll never know.
Best o' luck,