An external 32.768 kHz crystal with +/- 50 PPM accuracy or better should always be used for optimum functionality.
Some ANT capable devices include an internal 32.768 kHz oscillator with accuracy worse than +/-50 PPM. While this internal oscillator may appear to work under ideal, quiet wireless conditions for a single ANT channel, the channel will be far less robust in real world environments as compared to running with an external crystal.
An internal oscillator can be sensitive to temperature changes, so calibration would need to occur more often when the error spikes due to changes in the environment, causing timing issues and increased power consumption.
An error greater than +/-50PPM will mean that ANT channel transmissions could always fall outside of the normally expected timing window, which is conceptually reserved for ANT channel coexistence. While for a single ANT master channel in a quiet environment this would appear to work fine, this means that in noisy environments (or environments with many active ANT master channels like ANT+ in a training gym), the transmitter(s) would interfere with one another far more often, and cause issues with timing collisions over the air.
If the PPM happens to skew fully in opposite directions on the transmitting/receiving devices, then the channel may drop periodically as well, and search period durations could also be extended.
This ANT Application Note goes into some further discussion on the detrimental effects of poor clock accuracy, as well as a tool to help test the accuracy of your existing hardware.