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][1] 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.}}.  *)


  [1]: https://reference.wolfram.com/language/tutorial/NDSolveProjection.html