= Solution
All <gamma matrices> in this <basis> are real, and the <Lorentz transformation> $S(\theta)$ is real. Therefore <complex conjugation> commutes with the connected <Spinor representation of the Lorentz group>. With a suitable phase convention, <charge conjugation> is simply $\psi^c=\psi^*$. The <Majorana spinor> condition $\psi^c=\psi$ becomes
$$
\boxed{\psi_M=\begin{pmatrix}a\\b\end{pmatrix},\qquad a^*=a,\quad b^*=b.}
$$
The reality constraint is preserved by <Lorentz transformations> because $(S\psi_M)^*=S\psi_M$. For fermion fields, real components mean self-conjugate <Grassmann variables> or self-adjoint field components, rather than ordinary commuting real numbers.
Moreover, $\Gamma=\sigma_3$ is real, so <charge conjugation> commutes with the <chiral projectors>. The <Majorana spinor> and <Weyl spinor> conditions can therefore be imposed simultaneously. The <Majorana-Weyl spinors> are
$$
\boxed{\psi_{MR}=\begin{pmatrix}a\\0\end{pmatrix},\qquad
\psi_{ML}=\begin{pmatrix}0\\b\end{pmatrix},\qquad a^*=a,\ b^*=b.}
$$
The independent real component counts are
$$
\boxed{
\begin{array}{c|c}
\text{spinor type}&\text{real components}\\\hline
\text{Dirac}&4\\
\text{Weyl, either chirality}&2\\
\text{Majorana}&2\\
\text{Majorana-Weyl, either chirality}&1
\end{array}.}
$$
These are component counts before imposing an equation of motion. In particular the simultaneous reality and <chirality> constraints are compatible in this spacetime dimension, unlike the familiar four-dimensional case.
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