Yes, theremins with a “synchronization range”, where both frequencies coincide to a common frequency and no beat occurs, have a particular change in frequencies and timbre near this range:
When two coupled oscillators approach the synchronization region, their natural frequencies change with respect to the effects of coupling and detuning. The oscillators adjust their frequencies closer to each other, whereby the extent of this adjustment is influenced by the coupling strength.
If the detuning remains within the synchronization range, the oscillators synchronize to a common frequency. If the detuning exceeds this range, the oscillators oscillate more and more at their decoupled frequencies.
In the range, in which the oscillators "fight against each other" to synchronize the phases, the timbre also changes. Presumably, this change in the modulation waveform is not yet described with a mathematical background.
If the coupling is inductive and too strong, which was certainly the case with the old RCA theremin, the synchronization range can be partially or completely reduced with some capacitive coupling. I think this is how the Clara Rockmore RCA was improved by Lev Termen.
Next, I will post some simulation results for both effects.