How should I handle NDsolve in Manipulate?

Manipulate[
 sol = NDSolve[{a'[t] == -I*a[t] (delta + g*Re[b[t]]) - a[t]*0.04/2, 
    b'[t] == -I (b[t]*delta + g*Abs[a[t]]^2) - a[t]*0.09/2, a[0] == 1,
     b[0] == 1}, {a, b}, {t, 0, 200}];
 ParametricPlot[{Re[b'[t]], Re[b[t]]} /. sol, {t, 0, 200}, 
  PerformanceGoal -> "Quality"], {g, 0.000001, 
  0.00001}, {delta, -0.823 - 0.01, -0.823 + 0.01}]

You might consider the following variant of cvgmtj's answer. It has some performance advantages.

Manipulate[
  {aF, adF, bF, bdF} =
    NDSolveValue[
      {a'[t] == -I*a[t] (delta + g*Re[b[t]]) - a[t]*0.04/2, 
       b'[t] == -I (b[t]*delta + g*Abs[a[t]]^2) - a[t]*0.09/2, a[0] == 1,
       b[0] == 1}, 
     {a, a', b, b'}, {t, 0, 200}];
  ParametricPlot[{Re[bdF[t]], Re[bF[t]]}, {t, 0, 200}, 
    PerformanceGoal -> "Quality"],
  {aF, None},
  {adF, None},
  {bF, None},
  {bdF, None},
  {g, 0., 0.0005, Appearance -> "Labeled"},
  {delta, -0.823 - 0.25, -0.823 + 0.25, Appearance -> "Labeled"},
  TrackedSymbols :> {g, delta}]

I might not have posted this variant because the performance improvement is not all that noticeable, except that I want you to bring a further variant, which eliminates g, to your attention. I can see no visible difference in the plot produced from the following code form the plot produced by the preceding code. Can you?

Manipulate[
  {aF, adF, bF, bdF} =
    NDSolveValue[
      {a'[t] == -I*a[t] delta - a[t]*0.04/2, 
       b'[t] == -I b[t] delta - a[t]*0.09/2,
       a[0] == 1, b[0] == 1}, 
      {a, a', b, b'}, {t, 0, 200}];
  ParametricPlot[{Re[bdF[t]], Re[bF[t]]}, {t, 0, 200}, 
    PerformanceGoal -> "Quality"],
  {aF, None},
  {adF, None},
  {bF, None},
  {bdF, None},
  {delta, -0.823 - 0.25, -0.823 + 0.25, Appearance -> "Labeled"},
  TrackedSymbols :> {delta}]

Makes me wonder if g is actually a significant variable. Perhaps your mathematical model can be profitably simplified were you to eliminate g.

ClearAll[reParametricListLinePlot];

reParametricListLinePlot[
   ifs : {_InterpolatingFunction, _InterpolatingFunction}, 
   opts : OptionsPattern@ListLinePlot] := 
  ListLinePlot[Transpose[Re@#@"ValuesOnGrid" & /@ ifs], opts];

Manipulate[
 reParametricListLinePlot[
  NDSolveValue[
   {a'[t] == -I*a[t] (delta + g*Re[b[t]]) - a[t]*0.04/2,
    b'[t] == -I (b[t]*delta + g*Abs[a[t]]^2) - a[t]*0.09/2,
    a[0] == 1, b[0] == 1},
   {b', b}, {t, 0, 200}],
  InterpolationOrder -> 3, AspectRatio -> 1],
 {{delta, -0.823}, -2, -0.01, Appearance -> "Labeled"},
 {{g, 0.000005}, 0.000001, 0.0001, Appearance -> "Labeled"}
 ]