Let $(\mathcal{C},\otimes,\mathbb{I},\alpha,\lambda,\rho,c)$ be a braided monoidal category. By this I mean a (not necessarily strict) monoidal category $(\mathcal{C},\otimes,\mathbb{I},\alpha,\lambda,\rho)$ together with an invertible brading $$c_{X,Y}:X\otimes Y \to Y\otimes X.$$ In $\mathcal{C}$ we can consider two bialgebras $A$ and $B$ and their tensor product $A\otimes B$. It is well-known that if the braiding is symmetric (i.e. if $c^2=\mathsf{id}$) then $A\otimes B$ has a natural structure of bialgebra and it seems to me that it is folklore that in general this is not true. After long computations with braided diagrams, I convinced myself that one really needs the symmetry condition, whence now I would like to ask:
Question 1: Do there exist examples of bialgebras in braided monoidal categories whose tensor product is not a bialgebra anymore?
Question 2: Is there a trustful reference in which this is explicitly claimed?