  We establish a tannakian formalism of $p$-adic multiple polylogarithms and
$p$-adic multiple zeta values introduced in our previous paper via a comparison
isomorphism between a de Rham fundamental torsor and a rigid fundamental torsor
of the projective line minus three points and also discuss its Hodge and etale
analogues. As an application we give a way to erase log poles of $p$-adic
multiple polylogarithms and introduce overconvergent $p$-adic multiple
polylogarithms which might be $p$-adic multiple analogue of Zagier's
single-valued complex polylogarithms.
