While I agree that this passage gives a problematic description and appreciate user25930's answer, I think that the map you are using is not a particularly good one and that if you had a better map there might be a better explanation (maybe).
The biggest problem with the map you have (as concerns this case) is that it has Manasseh (both East and West) both running through the middle in one big impenetrable blob, but this is, I think, inaccurate.
The text, at Joshua 13:27 describes Gad as "having the Jordan as a boundary, to the lower end of the Sea of Chinnereth". So Gad's territory ran up the Jordan River to the sea of Chinnereth (Galilee), making a distinct break between the Manassehs. Your map also presumes that West Manasseh filled the space running South along the West side of the Jordan to Gilgal, but so far as I know, Joshua 16-17 doesn't describe this. In 17:7 it says that West Manasseh reached as far as Micmethath (the northernmost point of Ephraim), and verse 10 says that the Eastern border is Issachar (not the Jordan). So the West side of the Jordan river valley is left open and cartographers have little choice but to creatively guess at who got that area.
So, it is possible that there's a strip of land running down the west side of the Jordan which is unoccupied by any other tribe in the descriptions. I propose that Naphtali filled that narrow strip of land and reached down to Judah. I find some support for this in Deuteronomy 33:23, in which Moses blessed Naphtali, saying that they would possess "the lake and the south".
I compiled the map below using resources from BibleAtlas.org and many hours of other personal study. The tribal border outlines are my own:
An updated version of this map (along with interactive layers showing more information), can be found here.
You can see here that Naphtali is East of Judah. This works with the King James translation that Naphtali was "to Judah upon Jordan toward the sunrising"
The big problem with that proposal is that the text does say that both Issachar and Ephraim reached the Jordan (your map doesn't show that for Ephraim). If both of them touched the Jordan, it seems that Naptali wouldn't have had room to fit past them. I propose the following possible solutions:
Naphtali's territory was broken into pieces along that strip. You'll note that with many of the tribes, the borders aren't even really listed so much as the cities are listed. Our firm conception of strict borders doesn't really mesh with the ancient Hebrew idea. If Naphtalites were living in a place, it became part of Naphtali, whether or not it was all one contiguous land mass.
The borders of Ephraim and Issachar didn't so much reach the border as they had an easement though Naphtali to make use of the Jordan, because it was a valuable resource. My reading of Joshua 16 describes Ephraim that way, and you can see that on my map Ephraim has only a finger reaching down to meet the Jordan.
Ephraim and Issachar did reach the Jordan, but only in the sense that they reached major tributaries of the Jordan (as I have above for Issachar), giving them access to the river without blocking Naphtali's Southern strip.