Let us assume a simple rotation motion, when angular acceleration az somewhat depends on angle Phi[t] and set as linear interpolation function in range of 0..360 degrees. To not come out from interpolation range let angle Phi to be truncated to proper range by Mod[Phi][t], 360]:

az = Interpolation[{{0., 1.}, {90., 4}, {180., 7.}, {270., 4.}, {360.,
      1.}}, InterpolationOrder -> 1];

{\[Omega]s, \[CurlyPhi]s} = NDSolveValue[{

   \!\(
\*SubscriptBox[\(\[PartialD]\), \(t\)]\ \(\[Omega][
      t]\)\) == (az[Mod[\[CurlyPhi][t], 360]] + 10000),
   \!\(
\*SubscriptBox[\(\[PartialD]\), \(t\)]\ \(\[CurlyPhi][
      t]\)\) == \[Omega][t],

   \[CurlyPhi][0] == 359, \[Omega][0] == 
    390 }, {\[Omega], \[CurlyPhi]}, {t, 0, 0.0029}]

Let run it, and we get

InterpolatingFunction::dmval: Input value {360.052} lies outside the range of data in the interpolating function. Extrapolation will be used.

This is quite weird result - we have truncate phi by 360 and got 360.0520876770316`

Adding

EvaluationMonitor :> 
 Print[t, "  ", \[CurlyPhi][t], "  ", Mod[\[CurlyPhi][t], 360]]

Gives list like

...
0.00203059  359.813  359.813
0.00232059  359.932  359.932
InterpolatingFunction::dmval: Input value {360.052} lies outside the range of data in the interpolating function. Extrapolation will be used.
0.0026103  360.052  0.0520877
0.00248493  360.  360.
0.00248493  360.  0.
0.00248493  360.  0.
0.00248493  360.  2.19102*10^-6

Mod[] works fine in the EvaluationMonitor, but fails in main NDSolve loop.

Can anyone tell the reason of such behavior and how to fix it?