Restricting entries of unitary operators to real numbers and universal gate sets
There is A Simple Proof that Toffoli and Hadamard are Quantum Universal by Dorit Aharonov which first shows how complex amplitudes can be simulated by real amplitudes over a larger Hilbert space with one more qubit. 

"This is done by adding one extra qubit
to the circuit, the state of which indicates whether
the system’s state is in the real or imaginary part of
the Hilbert space, and replacing each complex gate
$U$ operating on $k$ qubits by its real version, denoted
$\tilde{U}$, which operates on the same $k$ qubits plus the
extra qubit. $\tilde{U}$ is defined by:

$\tilde{U}|i\rangle |0\rangle = [Re(U)|i\rangle]|0\rangle + [Im(U)|i\rangle]|1\rangle$
$\tilde{U}|i\rangle |1\rangle = -[Im(U)|i\rangle]|0\rangle + [Re(U)|i\rangle]|1\rangle$"

Secondly, she proofs the universality of the {Hadamard,Toffoli} gate set, which has only real amplitudes $\{0,1,\pm\frac{1}{\sqrt{2}}\}$.