Ticks and Frame don't play well together.
This get's you part of the way...

    Plot[
     With[{z = 1.1*E^(I u) + 0.908*E^(-I u)}, 
      Re[z] + 0.78*Log[Abs[z - 2]] + 0.64*Log[Abs[z - 1]] + 
       0.254*Log[Abs[z^2 - z - 1]] + 0.173*Log[Abs[z]]], {u, -15, 15},
     PlotStyle -> {Black},
     PlotRange -> {{-8, 8}, {-2, 1}},
     AspectRatio -> 1,
     AxesLabel -> (Style[#, 14] & /@ {"\[CurlyPhi]", "f"}),
     GridLines -> {Range[-8, 8, .5], Range[-2, 1, .2]},
     
     AxesOrigin -> {0, 0},
     Ticks -> {Range[-8, 8, .5], None},
     
     (* Frame ->True, *)
     (* FrameTicks-> {{Range[-2,1,.2],Automatic},None}, *)
     
     PlotLabel -> 
      Style[TraditionalForm[
        HoldForm[
         Re[z] + 0.78*ln[Abs[z - 2]] + 0.64*ln[Abs[z - 1]] + 
           0.254*ln[Abs[z^2 - z - 1]] + 0.173*ln[Abs[z]] <= 0.3999]], 
       "TR", Black, 14],
     ImageSize -> Large]

[![enter image description here][1]][1]

You could add a tick scale to the left side and show it with the plot in a `Row[ ]`

I'll give it some more thought.

  [1]: https://i.sstatic.net/cAjm0.png