Hmm, this is another miracle.

For me, your code executed in a freshly loaded kernel looks like:

After testing a bit more on another W10 computer I must correct myself. There the change from complex to real also appears, but later than it does on the Cloud. Typically around the 14th evaluation.

Table[{i, PlanarAngle[{0, 1} -> {{-Sin[\[Phi]], Cos[\[Phi]]}, {Cos[\[Phi]], Sin[\[Phi]]}}]}, {i, 50}]

Very very strange. Does anybody have any idea what happens here?

I think repeatability and reproducibility shoul be one of the most basic features of programming languages. PlanarAngle seems to show a kind of randomness instead.

@Jason: This makes results at least reproducible, just like Remove, ClearAll do (see above). But it is a very expensive operation since it clears all cached intermediate results so far.

We now have:

Of course it doesn't prevent PlanarAngle from changing its behavior.

I reported this as a bug to Wolfram Support.

@Hans: Yes, VectorAngle[u,v] remains stable. But VectorAngle is very different from PlanarAngle since it does not honor the direction of rotation. It gives the smaller of the two angles between u and v. Hence it is useless for calculating rotation angles. You must enhance it with quite complicated case diffentiations according to the relative positions of p, q1, q2. This is why PlanarAngle exists. See:

p = {0, 0}; q1 = {1, 0};
Print[{q2 = #, VectorAngle[q1 - p, q2 - p], 
     PlanarAngle[p -> {q1, q2}, "Counterclockwise"]}] & /@ {{-1, 
    1}, {-1, -1}};

{{-1,1},(3 \[Pi])/4,(3 \[Pi])/4}
{{-1,-1},(3 \[Pi])/4,(5 \[Pi])/4}

Your findings are touched upon in the PlanarAngle documentation