Sophie Turner Reactivates Divorced with Joe Jonas, Still Amicable

7:16 AM PT -- Joe's rep tells TMZ, "The filing was a legal formality, and the couple continues to negotiate an amicable resolution."

Sophie Turner and Joe Jonas have failed to hash out a settlement in their divorce, so now she's filed legal docs asking the judge to lay down the law and end the marriage.

Sophie's docs, obtained by TMZ, ask to "reactivate" the divorce proceedings, which were put on hold as the former couple went to mediation to work out a property and child custody settlement.

Apparently, those discussions broke down, so she's asking the judge to hear the case and impose a court-ordered settlement.

They went to mediation in part to keep their disputes private ... but that won't be the case now that she's asked to have a judge take it on ... all the filings -- outlining their disputes -- will now be public.

As we reported, Joe and Sophie had significant disagreements on where the children would live full-time ... she wanted to take them to the UK -- Joe wanted to keep them in the U.S. It got so bad, Sophie filed legal docs involving a child abduction law, but she dropped it.

It seems they're still at loggerheads over custody. They had a prenup, so property is probably not a big issue.

Joe and Sophie reached a temporary child custody arrangement, but it seems they've been unable to hash out a permanent one.

Both Joe and Sophie have moved on romantically ... he's with Stormi Bree, she's with Perry Pearson.

So it's game on ... again.