After reviewing this question as well as this one, I am left with some confusion, mainly about the nature of complex and real representations of the Lorentz group and how we do physics. I understand that representations are 'borrowed' from the complexification, and so naturally we consider how the Lorentz group acts over complex vector spaces. But then whenever possible we seem, via the following prescription, to descend to the real form of the group/algebra:
$$v = v_1+iv_2 \quad| \quad v_i\in\mathbb{R}^n$$ $$g\in SO(1,3;\mathbb{R})$$ $$\rho_\mathbb{C}(g)\cdot v = \rho_\mathbb{R}(g)\cdot v_1+i \rho_\mathbb{R}(g)\cdot v_2$$
My question is why everywhere in physics we choose to 'drop down' and work over real vector spaces except the spinor rep $(\frac{1}{2},0)\oplus(0,\frac{1}{2})$, when a real spinor rep does exist in the Majorana rep. Is this just convenience? Are there consequences if one chooses to work in a complex representation for, say, field strength tensors $(1,0)\oplus(0,1)$ (perhaps potentially unphysical/chiral field strength configurations)? Or is everything fine no matter what vector space we choose to work in?
