Wrong Eigenvalues result in Version 12.0

The following code calculates the eigenvalues of a certain complex matrix, which should come in pairs of opposite complex numbers. The code plots the real part of adding each pair. So the correct plot should be just zeros everywhere.

This is indeed the case in Version 10.1 & 11.3 as far as I tested. However, Version 12.0 gives something seriously wrong as shown below.

NN = 200; R = 0.05; xlist = Table[x, {x, -0.2 \[Pi], 0.2 \[Pi], 0.01}]; modl[n_] := 2*^-3 (Quotient[n, 2] - NN/2); t1 = -1 + Cos[x] - I Sin[x] + I R; t1p = -1 + Cos[x] + I Sin[x] + I R; t2a[n_] := -1 - modl[n]; t2b[n_] := -1 + modl[n]; mat[x_] =    DiagonalMatrix[     Table[If[EvenQ[n], t1, t2a[n]], {n, 0, 2 NN - 1 - 1}], 1] +     DiagonalMatrix[     Table[If[EvenQ[n], t1p, t2a[n]], {n, 0, 2 NN - 1 - 1}], -1] +     DiagonalMatrix[     Table[If[EvenQ[n], t2b[n], 0], {n, 0, 2 NN - 1 - 3}], 3] +     DiagonalMatrix[     Table[If[EvenQ[n], t2b[n], 0], {n, 0, 2 NN - 1 - 3}], -3]; list0 = Sort@Re@Eigenvalues[mat[xlist[[3]]]]; list0p = Table[list0[[i]] + list0[[2 NN - i + 1]], {i, NN}]; ListPlot[Tooltip@list0p, PlotRange -> All] 

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