Braided Hopf algebra, is a Hopf algebra object in braided category. In consequence, it has an usual bialgebra structure and satisfies braided-compatibility with braided antipode map.
In the case of ordinary Hopf algebra $H, K$ over $k$, there is a natural Hopf algebra structure on tensor product of $H, K$, with $m_{H \otimes K} := (m_H \otimes m_K) \circ (\mathrm{id} \otimes \mathrm{flip} \otimes \mathrm{id})$, $\Delta_{H \otimes K} : (\mathrm{id} \otimes \mathrm{flip} \otimes \mathrm{id}) (\Delta_H \otimes \Delta_K)$.
Generalize this for the case of braided Hopf algebra $M, N$. The possible candidates of new product $m_{M \otimes N}$ and coproduct $\Delta_{M \otimes N}$ are similar to the ordinary case, but $\mathrm{flip}$ is replaced by the braiding $\Psi$.
However, in this case, the braid-compaitibility part is really hard to check, and furthermore, it seems that it doesn't hold in general.
So, is there any way to define a natural braided Hopf algebra structure on the tensor product of two braided Hopf algebra? (Excepting the obvious, symmetric monoidal case)