How should I pre-process a real valued signal in order to use Kay's estimator?
The tool for converting real signals into their analytic representation is the Hilbert transform.
Suppose your signal were a projection of some helical rotation with variable amplitude onto the real-time plane, like in the image below.

Source
The Hilbert transform produces such complex signal given its real part. It is a linear transform and it's very easy to do in frequency domain. Without going into its mathematics and derivations too much in depth, the Fourier Transform imaginary part of your signal is the same as the one of your real signal multiplied by $-j$ (rotated by 90 degrees). By symmetrical properties of real signal, you get the following relationship:

All your negative ferquency components become 0.
Your DC component stays the same.
All your positive frequency components double

In Matlab, for example, you would do the following:
a = rand(1,201);

hilbert_a = ifft( [ 1, 2*ones(1,100), zeros(1,100)] .* fft(a) );

or simply use built-in the hilbert function.