Overlapping in real time fourier transform?
Overlapping FFTs, and averaging them, provide an excellent way to more accurately render the noise floor of the signal being observed. That said, you cannot actually start the FFT until you have enough samples, in part because the "last" sample will be used even on the very first butterfly computation.

That said, the time to compute the FFT can be quite short, and shortened further by putting part of it in an FPGA, depending on the signal that can achieve a very short time between the computation and the ability to act on that computation. The latency of that action would then be the number of bins in your FFT multiplied by the sample rate (time to get one FFT) plus the time to execute your recognition code and response. You can compare that time to the rate at which new signals appear or disappear in the sampled signal to see how close you are to "real time."

For audio data recording at 192K samples per second, each sample takes about 5 microseconds to come in, and even a small FPGA can compute a 1024 bin FFT in under a microsecond, so well before the sample after the last one has arrived.