  For the fifth Painlev\'e transcendents an asymptotic representation by the
Jacobi $\mathrm{sn}$-function is presented in cheese-like strips along generic
directions near the point at infinity. Its elliptic main part depends on a
single integration constant, which is the phase shift and is parametrised by
monodromy data for the associated isomonodromy deformation. In addition, under
a certain supposition, the error term is also expressed by an explicit
asymptotic formula, whose leading term is written in terms of integrals of the
$\mathrm{sn}$-function and the $\vartheta$-function, and contains the other
integration constant. This paper contains corrections of the Stokes graph and
of the related results in the early version.
