A recent question, Equality of electric charges of all leptons, made me wonder about a specific aspect of why exactly the charges of the different (free) fundamental particles are all the same. Specifically, why are the charges of the different (charged) leptons all equal?

I understand this is a requirement in any treatment that includes the weak force, since the existence of $W^-\to e^-\bar{\nu}_e$, $W^-\to \mu^-\bar{\nu}_\mu$ and $W^-\to \tau^-\bar{\nu}_\tau$ vertices requires all the charges present to be equal. These vertices enable decays of the form $\mu^-\to e^-\bar{\nu}_e\nu_\mu$, which can only conserve charge if there's a single lepton charge unit.

If you take away this interaction, though, and you simply consider a multi-species QED theory, this mechanism goes away, and it would seem that you could in principle have different charges for the different species. Is this consistent with QED? Or is there something else going on, within QED, which requires the charges to be equal?