Trying to provide a short-winded answer: The chiral projection operator $$ P_{RL} = \frac{1}{2}(1 \pm \gamma^5), $$ involves $\gamma^5$, which is expressed as $$ \gamma^5 = i\gamma^0\gamma^1\gamma^2\gamma^3. $$ The imaginary number $i$ in above definition is crucial in keeping $\gamma^5$ hermitian $$ (\gamma^5)^\dagger = \gamma^5. $$ Given that the Standard model is chiral, the indispensable $i$ in the definition of $\gamma^5$ (and thus chiral projection $P_{RL}$) behooves us to choose a complex representation.

That being said, a real representation is not strictly prohibited if you are innovative enough to come up with real chiral projection operator and real $\gamma^\mu$ representation.

Trying to provide a short-winded answer: The Standard Model is chiral, and we define the chiral projection operator as $$ P_{RL} = \frac{1}{2}(1 \pm \gamma^5), $$ which involves $\gamma^5$ expressed as $$ \gamma^5 = i\gamma^0\gamma^1\gamma^2\gamma^3. $$ The imaginary number $i$ in above definition is crucial in keeping $\gamma^5$ hermitian $$ (\gamma^5)^\dagger = \gamma^5. $$ Given that the Standard model is chiral, the indispensable $i$ in the definition of chiral projection $P_{RL}$ behooves us to choose a complex representation.

That being said, a real representation is not strictly prohibited if you are innovative enough to come up with real chiral projection operator and real $\gamma^\mu$ representation.

Trying to provide a short-winded answer: The chiral projection operator $$ P_{RL} = \frac{1}{2}(1 \pm \gamma^5), $$ involves $\gamma^5$, which is expressed as $$ \gamma^5 = i\gamma^0\gamma^1\gamma^2\gamma^3. $$ The imaginary number $i$ in above definition is crucial in keeping $\gamma^5$ hermitian $$ (\gamma^5)^\dagger = \gamma^5. $$ Given that the Standard model is chiral, the indispensable $i$ behooves us to choose a complex representation.

That being said, a real representation is not strictly prohibited if you are innovative enough to come up with real chiral projection operator and real $\gamma^\mu$ representation.

Trying to provide a short-winded answer: The chiral projection operator $$ P_{RL} = \frac{1}{2}(1 \pm \gamma^5), $$ involves $\gamma^5$, which is expressed as $$ \gamma^5 = i\gamma^0\gamma^1\gamma^2\gamma^3. $$ The imaginary number $i$ in above definition is crucial in keeping $\gamma^5$ hermitian $$ (\gamma^5)^\dagger = \gamma^5. $$ Given that the Standard model is chiral, the indispensable $i$ in the definition of $\gamma^5$ (and thus chiral projection $P_{RL}$) behooves us to choose a complex representation.

That being said, a real representation is not strictly prohibited if you are innovative enough to come up with real chiral projection operator and real $\gamma^\mu$ representation.