Yes, it is consistent with AD. In $L(\mathbb{R})$, if AD holds then $\Theta$ injects into $\mathcal{P}(\mathbb{R})$, or equivalently, into $2^{\mathbb{R}}$. The OD sets of reals are Wadge-cofinal and $\Theta$ is regular, so we can define an injection $F : \Theta \to \mathcal{P}(\mathbb{R})$ by induction. Let $F(\alpha)$ be the $<_{OD}$-least OD set of reals with Wadge rank greater than the Wadge ranks of all the sets of reals $F(\gamma)$, $\gamma < \alpha$.
To see that in $L(\mathbb{R})$ the Wadge ranks of the OD sets of reals are cofinal, notice that every set of reals $A$ is OD from some real $y$, say $x \in A \iff \varphi[x,y,\alpha]$, and the OD set $\lbrace (x,y) : \varphi[x,y,\alpha] \rbrace$ has Wadge rank at least that of $A$.
Assuming $AD + V=L(\mathcal{P}(\mathbb{R}))$, the existence of an injection $\Theta \to \mathcal{P}(\mathbb{R})$ is equivalent to the length of the Solovay sequence being a successor ordinal. The meaning of this expression that we will use here is that there is a set of reals $A$ such that the $OD_A$ sets of reals are Wadge-cofinal in $\mathcal{P}(\mathbb{R})$. This implies that $\Theta$ is regular, so we just modify the argument given for $L(\mathbb{R})$ by adding $A$ as a parameter.
Conversely, if there is an injection $F:\Theta \to \mathcal{P}(\mathbb{R})$, it must be Wadge-cofinal or else we would get an injection $\Theta \to \mathbb{R}$, which is impossible. Because $V = L(\mathcal{P}(\mathbb{R}))$, the function $F$ is OD from a set $A$ of reals, so its range is a Wadge-cofinal collection of $OD_A$ sets of reals.