Replacing one IC seems to work:
ClearAll[f, F, inv, sol]
f[w_?NumericQ] := Im[PolyLog[2, -E^(I*w)]]
F[w_?NumericQ] := NIntegrate[f[t], {t, 0, w}]
inv[w_?NumericQ, wp_?NumericQ]:= 0.5*wp^2+F[w]
sol[l_, c_] := NDSolve[
{wp[x] == w'[x], wp'[x] == -f[w[x]], wp[ 0] == c, Mod[Abs[w[0] - w[l]], 2*Pi] == 0},
{w, wp},
{x, 0, l},
MaxStepFraction -> 0.01,
MaxSteps -> Infinity,
Method -> {"FixedStep", Method -> Automatic}
];
out = sol[5, 0.2] // First ;
w[5] - w[0] /. out
{wp[0], wp[5]} /. out
F[w[5] /. out] - F[w[0] /. out]
Table[inv[w[x], wp[x]] /. out, {x, 0, 5, 0.5}]
I'd expect projection method to work here, but for some reason it throws an error:
sol[l_, c_] := NDSolve[
{wp[x] == w'[x], wp'[x] == -f[w[x]], wp[ 0] == c, wp[l] == c},
{w, wp},
{x, 0, l},
MaxStepFraction -> 0.01,
MaxSteps -> Infinity,
Method -> {"FixedStep", Method -> {"Projection", Method -> Automatic, "Invariants" ->{inv[w[x], wp[x]] }}}
];
out = sol[5, 0.2] // First ;
(* NDSolve::nnum1: The function value inv[w[0.],wp[0.]] is not a number when the arguments are {0.,{0.,0.}}. *)