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