Did Confederate and Union soldiers fraternize during the Civil War?

After a fashion.

For one thng, there were masses of desertions from one army to another, or away from the armies altogether, where solders from both sides would go on the run together.

In the Siege of Vicksburg, Union gunners would shout a warning to 'Johnny Reb' to get his head down before they blew up the town's defences.

In the winter retreat of 1862-3, the two armies were camped on either side of the Rappahannock, and when the Union army rehearsed the band, the Confederates would sing along with it. (The bandmaster even played requests like 'Dixie'.)