As a teenager, when examining an equinox-1950 star atlas, I noticed that the galactic equator crossed the ecliptic very close to the latter’s northern and southern extremes (that is, the solstices at right ascension 6h and 18h). On equinox-2000 charts they are even closer. I’d love to know when the exact coincidence will (or did) occur.
You are not alone. Jean Meeus addresses this very question in chapter 48 of his 1997 book Mathematical Astronomy Morsels. Because of precession, the ecliptic longitude of each intersection point increases 50.3 arcseconds per year. He finds that the galactic equator (as defined by the International Astronomical Union in 1959) passed over the solstice points during May 1998.
— Roger W. Sinnot