Just ignore the gaps. It's not like you can actually fall into them, so let's have a disco-ball world.
This is how the world map might be. A meridian line for reference (north-south travels directly up/down it), the rest #s where north/south is increasingly disjointed. The arrows indicate "go to the other side for west from west edge or east from east edge" and north/south from any leftmost or rightmost edge sends you to the leftmost/rightmost of the next row (however many columns it is away) with other non-meridian norths/souths being proportonally between the two.
The N and S are polar world-tiles which are respectively North/South from
any of the adjacent row of eight (nicely tuned to be eight tiles so that the eight cardinal/and semicardinal directions from the pole send you to just
one neighbour-latitude tile. But other attempts to go (say) 1N,1S might round you back onto a different tile from where you started this double-hop.
N
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S
(That's mathematically tuned to 10° of latitude per tile but slightly under 8° of longitude per tile on the equator (the easiest way to end up with the convenient 8 pole-adjacent tiles!), increasing as the absolute latitude increases. And rounded, so really there's fractional tiles missing/extra to get them to meet exactly half way round in an integer number (gaps and overlaps, shared around the ring). No easy way tune that out without using dissimilarly-sized tiles according to latitude.)
Obviously a real-world would have more than 479 tiles and not take up the entire world. This is just a zoom-out. And if you were cursoring around the globe you'd probably spin the latitudes around (pop #s off one side, push them onto the other, at the right time to make sense to keep the |-row vertical) so that direct north-south lies upon the meridian at all times and it's only distorted at the new fringes.
Zoomed in, you'd see a subset of the map, which could be rendered 'flatter' and with less obvious disjoints (you may still happen upon 1N+1S≠OriginalPosition problem, periodically) until you start to get quite close to the pole. At and over the pole you get
far more than eight neighbouring tiles around the pole (technically). So maybe falling back upon the 'pumped-up cube' model to map polar regions in a better grid format, as already discussed.
Also check out alternate
map projections with various solutions/new-problems to mapping your grid (discoball or otherwise) to the sphere.
(Slightly ninjaed?)