Recall that a super Reinhardt cardinal $κ$, is a cardinal which is the critical point of elementary embeddings $j:V→V$, with $j(κ)$ as large as desired.
Claim. If $\kappa$ is super Reinhardt, then tree property holds for a class of cardinals.
First note that it is easily seen that if $\kappa$ is as above, then $\{\alpha < \kappa: \alpha$ is supercompact$ \}$ is unbounded in $\kappa.$ It follows that $\{ \alpha < \kappa: TP(\alpha) \}$ is unbounded in $\kappa,$ where $TP(\alpha)$ is the assertion ``Tree property holds at $\alpha$''. The point is the Magidor-Shelah theorem that tree property holds at successor of singular limits of supercompact cardinals (one has to check their proof works without the use of AC and using the definition of supercompact cardinals given in the non-AC case).
It is now evident that $\{ \alpha: TP(\alpha) \}$ should be a proper class: if not, let $\lambda$ be a bound, and let $j: V \to V$ be such that $crit(j)=\kappa$ and $j(\kappa) > \lambda^+.$ As $\{ \alpha < \kappa: TP(\alpha) \}$ is unbounded in $\kappa,$ by elementarity $\{ \alpha < j(\kappa): TP(\alpha) \}$ is unbounded in $j(\kappa),$ which is in contradiction with the choice of $\lambda.$
Remark. The same idea seems to work for extendible cardinals or even a storng limit of supercompact cardinals $\kappa.$