Is it possible to turn off Construct's sub-pixel calculations? Should I just round the values every tick for the relevant objects?
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If you mean for rendering, then the project Properties bar does support Pixel rounding (change from Off to On). However, for calculations I believe you'll have to use rounding via ceil(your value), floor(your value), or round(your value).
Does this have any side-effects? For example differences in different resolutions?
The pixel rounding at render? I don't believe so, all it does is stop things from rendering at sub-pixel amounts by rounding to nearest whole pixel.