Interpolation in Contrast Limited Adaptive Histogram Equalization
In step 8 of the reference you provide, you obtain 4 possible values for a given pixel (based on the values obtained from its 4 neighbours, which I assume are the usual North-East-South_West neighbours).
Then, to avoid introducing tile artifacts, you can use any sufficiently regular interpolation method to obtain the value in the center of these 4 pixels.

For simplicity and computational issues, I suggest that you try bilinear interpolation. It will not introduce ringing or ripple artifacts and will not overflow the dynamic range of the input pixels.