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.}}.  *)

**Edit**

    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]
    
    ClearAll[solution] ;
    Options[solution] = {MaxStepFraction -> 0.005, MaxSteps -> Infinity, 
       Method -> {"FixedStep", 
         Method -> {"ImplicitRungeKutta", "DifferenceOrder" -> 10}}} ;
    solution[l_, c_,  opts : OptionsPattern[]] := 
      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}, opts] /; 
       c >= 0 ;
    solution[l_, c_,  opts : OptionsPattern[]] := 
      NDSolve[{wp[x] == w'[x], wp'[x] == -f[w[x]], wp[l] == c, 
         Mod[Abs[w[0] - w[l]], 2*Pi] == 0}, {w, wp}, {x, 0, l}, opts] /; 
       c < 0 ;
    
    out = solution[5, 0.2] // First;
    {wp[0], wp[5]} /. out
    Table[inv[w[x], wp[x]] /. out, {x, 0, 5, 1}]
    (* {0.2`,0.19999537137167114`} *)
    (* \
    {-1.6718455006855417`,-1.6718458095149678`,-1.671842921367624`,-1.\
    6718458808125778`,-1.6718464107564595`,-1.6718464264002912`} *)
    
    out = solution[5, -0.2] // First;
    {wp[0], wp[5]} /. out
    Table[inv[w[x], wp[x]] /. out, {x, 0, 5, 1}]
    (* {-0.19999537137418164`,-0.2000000000000036`} *)
    (* \
    {-1.6718464264011181`,-1.671846410757299`,-1.6718458808139036`,-1.\
    6718429213688029`,-1.671845809516107`,-1.67184550068668`} *)

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