Your plotfunction is   real if both Log-arguments are `>0`.
Perhaps using RegionFunctions might help solving your problem:

    Plot[ Log[27, Sin[2*x] - 1/3*Cos[x]] -1/3*Log[3, -Cos[x]] , {x, -2 Pi, 2 Pi}, PlotRange -> All, 
    RegionFunction ->Function[x, (Sin[2*x] - 1/3*Cos[x] > 0) && ( -Cos[x] > 0)],PlotRange -> All, PlotLabel -> "both Log-arguments>0"]

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

**addition**

But the sum of two comlex numbers might although evaluate to real, if the summands are complex:

    Plot[{ #, Im[#]} &[Log[27, Sin[2*x] - 1/3*Cos[x]] - 1/3*Log[3, -Cos[x]]] // Evaluate, {x, -2 Pi, 2 Pi}, PlotStyle -> {{ Red}, Green }]
    

[![enter image description here][2]][2]

and that's the result MMA calculates in the first plot of the question.

By the way it's identical to the use of `RegionFunction -> Function[{x, y},  Im[y] == 0] `
  
    Plot[ Log[27, Sin[2*x] - 1/3*Cos[x]] -1/3*Log[3, -Cos[x]] , {x, -2 Pi, 2 Pi}, PlotRange -> All, 
    RegionFunction -> Function[{x, y},  Im[y] == 0], PlotRange -> All, PlotLabel -> "Im==0"]

  [1]: https://i.sstatic.net/S85FM.jpg
  [2]: https://i.sstatic.net/0qVD2.jpg