It is a theorem of $\text{ZF+DC}$ that if $T$ is a weakly homogeneous tree on $\omega\times \kappa$ some $\kappa$ with homogeneity system of measures $\vec{\mu}$ and $ms(T,\vec{\mu})$ is the Martin-Solovay tree associated to $T$ then $p[T]$ and $p[ms(T,\vec{\mu})]$ are complements. See Cabal reprints volume 1 Jackson's article theorem 4.10 for a proof. Note that to show homogeneity of trees on $\omega\times \kappa$ for $\kappa<\Theta$ one actually only needs $\text{AD}$ (result of Woodin).

$\text{AD}$ or $\text{AD}_{\mathbb{R}}$ are used to show homogeneity or weak homogeneity of trees. This is done using partition properties in the determinacy context (see Jackson's article mentionned above). These assumptions are not needed to establish that a weakly homogeneous tree $T$ and its Martin-Solovay twin project to complements (if that's what you were looking for).

It is a theorem of $\text{ZF+DC}$ that if $T$ is a weakly homogeneous tree on $\omega\times \kappa$ some $\kappa$ with homogeneity system of measures $\vec{\mu}$ and $ms(T,\vec{\mu})$ is the Martin-Solovay tree associated to $T$ then $p[T]$ and $p[ms(T,\vec{\mu})]$ are complements. See Cabal reprints volume 1 Jackson's article theorem 4.10 for a proof. Note that to show weak homogeneity of trees on $\omega\times \kappa$, where $\kappa$ less than the supremum of the Suslin cardinals, one actually only needs $\text{AD}$ (result of Woodin).

$\text{AD}$ or $\text{AD}_{\mathbb{R}}$ are used to show homogeneity or weak homogeneity of trees. This is done using partition properties in the determinacy context (see Jackson's article mentionned above). These assumptions are not needed to establish that a weakly homogeneous tree $T$ and its Martin-Solovay twin project to complements (if that's what you were looking for).