Voltage controlled phase shift
You can use a phase locked loop chip to create a version of your input voltage that is 90 degrees distant.

If you used a potentiometer with top connected to original and bottom connected to phase shifted version then the wiper can sweep an output that is phase shifted anywhere between zero and 90 degrees.

Imagine that pot was a digipot then you get digital control of the phase angle. If you wanted analogue control then use an ADC to digitize the control signal.

You can also use an opamp integrator to generate a 90 degree signal and this may be preferable in terms of simplicity.

Making this work across the full 360 can be done by using an opamp inverter to add or subtract 180 degrees to the original and the 90 degree signals.

You could also use comparators and analogue switches to make the process fairly seamless.