Is real-time Hilbert envelope feasible?
As with most things in Engineering the answer is "depends on your requirements". In general you can't do a perfect Hilbert transform unless the signal is strictly periodic (see http://andrewduncan.net/air/ ). Since you can't do perfect you need to define what's good enough for your particular application.

For most typically audio applications, that it's pretty straight forward. The Hilbert transform can be represented as a simple linear time invariant filter. Unfortunately the impulse response infinite and non-causal. For an FIR implementation you need to truncate and delay it. The larger your truncation window and your delay the more precise it will be, however it will also become more expensive computationally and the latency goes up. 

If your application requires a really long filter (typically when you need high precision at very low frequencies) than FFT can be useful in an overlap add configuration.

If you just care about relative phase and not absolute phase you can run the signal through a set of differential allpass filters, which is a very efficient way of implementing a band limited 90 degree phase shift.

You can also use advanced filter design methods (least square, Parks McLellan etc.) to optimize any Hilbert approximation for your specific requirements.