  Real stabilizer operators, which are also known as real Clifford operators,
are generated, through composition and tensor product, by the Hadamard gate,
the Pauli Z gate, and the controlled-Z gate. We introduce a normal form for
real stabilizer circuits and show that every real stabilizer operator admits a
unique normal form. Moreover, we give a finite set of relations that suffice to
rewrite any real stabilizer circuit to its normal form.
