How to produce 6 PWM signals with 60 degrees phase shift using Arduino Uno 6 PWM pins like in the image below?
Using hardware it's not possible. Simply because you effectively only have three PWM timer units, each with two compare outputs.
That means you have three timebases, which are each compared with two values for two PWM signals each.
You would need a single master timebase with six independent offsettable counters with comparison for your needs.
It would be possible to (roughly) do it in software, though, depending on your needs. You get jitter from software generated PWM, due to interrupts getting in the way of your smooth execution. However you have complete control over how the waveform is generated. Instead of a 0-X duty cycle you could program it to have an X-Y duty cycle. And with all of them taken from the same counter value it's then all synchronised 100%.