In an elliptically polarized antenna, the electric field along the x-direction is different from the electric field in the y-direction. The phase difference between these two waves is other than 90 degrees.
It depends on the isolation you require between channels. Circular polarization is used to give rain reject (a single reflection reverses the direction of circular polarization) or to give isolation between two channels of opposite polarization.
It is the amount of contamination from the opposite polarization that you can tolerate that sets the limit to the phase difference and the amplitude difference you can accept. That depends on the specification of the system. It might be -30 dB.
The relationship between the ratio of the two circular polarizations in the signal, and the phase difference and the amplitude difference between the two linear components can be worked out using simple geometry. It will be described in any standard textbook treating circular polarization, and maybe even wikipedia.
Circular polarization is also used so that it doesn't matter which way up the two antennas are, because the signal strength stays constant if one or both of the antennas rotate, such as a spinning missile.
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