A complex-linear operator comes from a real operator exactly when it commutes with the chosen conjugation

If $T=\theta \circ S_{\mathbb{C}}\circ {\theta }^{-1}$ for a real-linear $S$, then $T$ commutes with $\sigma$: for $w=\theta (z\otimes v)$, one has $T\sigma w=\theta S_{\mathbb{C}}{\theta }^{-1}\theta {\sigma }_{\mathrm{can}}(z\otimes v)=\theta S_{\mathbb{C}}(\bar{z}\otimes v)=\theta (\bar{z}\otimes Sv)$, while $\sigma Tw=\theta {\sigma }_{\mathrm{can}}{\theta }^{-1}\theta (z\otimes Sv)=\theta (\bar{z}\otimes Sv)$ by [L1] and [L2].

Conversely, if $T\sigma =\sigma T$, then $V$ is invariant under $T$: for $v\in V$, $\sigma (Tv)=T\sigma v=Tv$, so $Tv\in V$ by [L3]; the restriction $S:=T{\mathrm{\mid }}_V:V\to V$ is therefore a well-defined real-linear operator.

With this $S$, one has $\theta \circ S_{\mathbb{C}}\circ {\theta }^{-1}=T$: for $w=\theta (z\otimes v)$, $\theta S_{\mathbb{C}}(z\otimes v)=\theta (z\otimes Sv)=z\cdot Sv=T(zv)=T(\theta (z\otimes v))=Tw$, using [L1], the complex-linearity of $T$, and the identity $\theta (1\otimes v)=v$ of [L4].

Uniqueness: if $T=\theta \circ S_{\mathbb{C}}\circ {\theta }^{-1}=\theta \circ R_{\mathbb{C}}\circ {\theta }^{-1}$, then $S_{\mathbb{C}}=R_{\mathbb{C}}$ because $\theta$ is an isomorphism, and evaluating on $1\otimes v$ gives $\iota Sv=S_{\mathbb{C}}(1\otimes v)=R_{\mathbb{C}}(1\otimes v)=\iota Rv$, whence $Sv=Rv$ by the injectivity of the embedding in [L4].

Steps 1.1, 2.1 and 3.1 together prove both directions of the claimed equivalence, the concrete description of $S$ as the restriction, and its uniqueness.