How to eliminate extraneous lines when using Mod within ParametricPlot?

I’ve found two other similar questions:

Plot of the Mod of NDSolve solutions sometimes connects discontinuous jumps

and

How can I remove horizontal lines on a phase plot for a pendulum?

Neither contains a solution to my problem, but in an answer to the first there is a tantalizing note: This seems to have been fixed in V11. Alas I’m running V11 and getting a very similar issue that I would love to get rid of. Here’s my code:

Manipulate[
 Animate[sol = 
   NDSolve[{x'[t] == w1 + k1*Sin[y[t] - x[t]], 
     y'[t] == w2 + k2*Sin[x[t] - y[t]], x[0] == x0, y[0] == y0}, {x, 
     y}, {t, 0, tmax}];
  ParametricPlot[Mod[{x[t], y[t]} /. sol, 2 Pi], {t, 0, tplot}, 
   PlotRange -> {{0, 2 Pi}, {0, 2 Pi}}], {tplot, 0, tmax}],
 {{w1, 1}, -2, 2}, {{w2, 1}, -2, 2}, {{k1, 0.1}, 0, 2}, {{k2, 0.1}, 0,
   2}, {{x0, 0}, 0, 2 Pi}, {{y0, 0.2}, 0, 2 Pi}, {{tmax, 10}, 0, 100}
 ]

This is a simple coupled oscillator problem; the phase space is naturally the torus, i.e. the square with opposite edges identified. The obvious way to plot the solution then is Mod[], but we get the following:

The horizontal and vertical line shouldn’t be there. I understand why they’re added by ParametricPlot but I would like to get rid of them. I have tried using Exclusions; I have tried using WhenEvent within the NDSolve to reset x and y to zero so that the Mod[] is unnecessary and I get exactly the same output. Surely there’s a solution? Perhaps ParametricPlot can be told to ignore jumps greater than $varepsilon$? I don’t need it to be perfect.

Manipulate[Animate[sol = NDSolveValue[{x'[t] == w1 + k1*Sin[y[t] - x[t]], 
     y'[t] == w2 + k2*Sin[x[t] - y[t]], x[0] == x0, y[0] == y0}, {x,  y}, {t, 0, tmax}];
  ParametricPlot[Mod[Through@sol@t, 2 Pi], {t, 0, tplot}, 
   PlotRange -> {{0, 2 Pi}, {0, 2 Pi}}, PerformanceGoal -> "Quality", 
   ImageSize -> 300], {tplot, 0, tmax}, Alignment -> Center], 
{{w1, 1}, -2, 2}, {{w2, 1}, -2, 2}, {{k1, 0.1}, 0, 2}, {{k2, 0.1}, 0, 2}, 
{{x0, 0}, 0, 2 Pi}, {{y0, 0.2}, 0, 2 Pi}, {{tmax, 10}, 0, 100}, 
Alignment -> Center, ContentSize -> 500]

Manipulate[Animate[sol =  NDSolve[{x'[t] == w1 + k1*Sin[y[t] - x[t]], 
     y'[t] == w2 + k2*Sin[x[t] - y[t]], x[0] == x0, y[0] == y0}, {x, y}, {t, 0, tmax}];
  m[t_] := Mod[{x[t], y[t]} /. sol, 2 Pi]; 
  ParametricPlot[m[t], {t, 0, tplot}, PlotRange -> {{0, 2 Pi}, {0, 2 Pi}}, 
   PerformanceGoal -> "Quality"], {tplot, 0, tmax}], 
 {{w1, 1}, -2, 2}, {{w2, 1}, -2, 2}, {{k1, 0.1}, 0, 2}, {{k2, 0.1}, 0, 2}, 
 {{x0, 0}, 0, 2 Pi}, {{y0, 0.2}, 0, 2 Pi}, {{tmax, 10}, 0, 100}]

Manipulate[
 sol = NDSolve[{x'[t] == w1 + k1*Sin[y[t] - x[t]], 
    y'[t] == w2 + k2*Sin[x[t] - y[t]], x[0] == x0, y[0] == y0}, {x, 
    y}, {t, 0, tmax}]; Animate[
  ListPlot[
   Table[Mod[{x[t], y[t]} /. sol, 2 Pi], {t, 0, tplot, 0.05}], 
   PlotRange -> {{0, 2 Pi}, {0, 2 Pi}}, 
   PlotStyle -> {Blue, PointSize[Tiny]}],
  {tplot, 0, tmax}], {{w1, 1}, -2, 2}, {{w2, 1}, -2, 2}, {{k1, 0.1}, 
  0, 2}, {{k2, 0.1}, 0, 2}, {{x0, 0}, 0, 2 Pi}, {{y0, 0.2}, 0, 
  2 Pi}, {{tmax, 10}, 0, 100}]