= Solution
For $x'=\Lambda x$, a <Dirac spinor> and vector field transform as
$$
\psi'(x')=S(\Lambda)\psi(x),
\qquad
\bar\psi'(x')=\bar\psi(x)S(\Lambda)^{-1},
\qquad
A'_\mu(x')=\Lambda_\mu{}^\nu A_\nu(x),
$$
where $S^{-1}\gamma^\mu S=\Lambda^\mu{}_\nu\gamma^\nu$. Hence the spinor terms and $F_{\mu\nu}F^{\mu\nu}$ are <Lorentz scalar>[Lorentz scalars]. The Maxwell term is real. Integration by parts and $(\gamma^\mu)^\dagger=\gamma^0\gamma^\mu\gamma^0$ show that the adjoint of $i\bar\psi\gamma^\mu D_\mu\psi$ differs from it by $-i\partial_\mu(\bar\psi\gamma^\mu\psi)$. Finally, $i[\gamma^\mu,\gamma^\nu]$ is Dirac-Hermitian, so the real coefficient $\lambda$ makes the Pauli bilinear real. Thus the action is real.
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