  In \cite{IKLNDP23} we presented a systematic study of algebra structures of
multiple zeta values in positive characteristic introduced by Thakur as
analogues of classical multiple zeta values of Euler. In this paper we
construct algebra and Hopf algebra structures of alternating multiple zeta
values introduced by Harada, extending our previous work. Our results could be
considered as an analogue of those of Hoffman \cite{Hof00} and Racinet
\cite{Rac02} in the classical setting. The proof is based on two new
ingredients: the first one is a direct and explicit construction of the shuffle
Hopf algebra structure, and the second one is the notion of horizontal maps.
