  This paper proposes a real moment-HSOS hierarchy for complex polynomial
optimization problems with real coefficients. We show that this hierarchy
provides the same sequence of lower bounds as the complex analogue, yet is much
cheaper to solve. In addition, we prove that global optimality is achieved when
the ranks of the moment matrix and certain submatrix equal two in case that a
sphere constraint is present, and as a consequence, the complex polynomial
optimization problem has either two real optimal solutions or a pair of
conjugate optimal solutions. A simple procedure for extracting a pair of
conjugate optimal solutions is given in the latter case. Various numerical
examples are presented to demonstrate the efficiency of this new hierarchy, and
an application to polyphase code design is also provided.
