Time response of complex frequency response function
A real signal with a phase shift does not technically exist since phase is $e^{j\phi}$, so requires an imaginary component except for phases that are modulo $\pi$. (The phase of a real signal is $0$ or $\pi$ radians only). You may be actually looking for a time delay. A constant time delay has a linear phase with frequency, and for the result to be real in time, the frequency must be complex conjugate symmetric. So to create a real only time delay use a linear phase in frequency that is complex conjugate symmetric such as the plot below.



If the delay in time need not be the same for all frequency components of the signal, then the phase response in frequency need not be linear (resulting in group delay variation)- but to be real in time the requirement for complex conjugate symmetry still holds.    

See my related "DSP Puzzle" focusing on this confusion of phase and delay: $9\rm V$ Battery with $45^\circ$ phase