Generate random numbers with specific Fourier spectrum

I have a set of random numbers that will be used for a simulation. However, I need this numbers to have a specific Fourier spectrum (that looks similar to the real data I have) but without changing the phase of the random numbers. Does anyone have any idea on how I can use the amplitude of the Fourier transform of the real data to generate approximately similar Fourier spectrum for the random numbers?

What I thought of doing is:

1) Take the Fourier transform of the real data.

2) Multiply the spectrum (|F(w)|) of the real data by the Fourier transform of the random numbers.

3) Calculate the inverse Fourier transform of the multiplied signal to get the random numbers.

Would this approach work well? What would be the effect on the phase angle (if any)? Any suggestions on different ways to do that are welcome.

Thank you very much