I always have problems with tuning and intonation on the third string, especially between the open G natural and the 1st fret G#. Now, part of this may be the consequence of tuning in equal temperament, in which thirds are always a problem, but I think it goes beyond that.
When I look at a standard set of strings, the G string looks much heavier than the B, which is only a major third away in pitch – 4/5. Why is this? I once had a set of strings that had a wrapped G string (silk wrapped in silver!) but I was too inexperienced to notice whether that made a difference. Folk guitar strings (metal) typically have the third string wrapped.
I am a newbie here — if this is too technical for this forum, please suggest a cross-posting. But I think nearly everybody, amateur or professional, must have wrestled with the difficulties and compromises of tuning.