  In this paper we study an analog of minimal surfaces called Weyl-minimal
surfaces in conformal manifolds with a Weyl connection $(M^4,c,D)$. We show
that there is an Eells-Salamon type correspondence between nonvertical
$\mathcal{J}$-holomorphic curves in the weightless twistor space and branched
Weyl-minimal surfaces. When $(M,c,J)$ is conformally almost-Hermitian, there is
a canonical Weyl connection. We show that for the canonical Weyl connection,
branched Weyl-minimal surfaces satisfy the adjunction inequality
\begin{equation}\label{adj} \chi(T_f\Sigma)+\chi(N_f\Sigma) \le \pm
c_1(f^*T^{(1,0)}M). \end{equation} The $\pm J$-holomorphic curves are
automatically Weyl-minimal and satisfy the corresponding equality.
