  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. Instead of the justification scheme for asymptotic
solutions of Riemann-Hilbert problems by the Brouwer fixed point theorem, we
begin with a boundedness property of a Lagrangian function, which enables us to
determine the modulus of the $\mathrm{sn}$-function satisfying the Boutroux
equations and to construct deductively the elliptic representation.
