Both dogs hear a gate open and sprint to the same line. They bark, run back and forth, and part only when someone calls one indoors. Tomorrow the fence is still there, and so is the pattern.
Start with safety. Check for gaps, loose panels, and places where a muzzle or paw can reach through. Supervise yard time and bring your dog in before both dogs get wound up. Don't put your hand between the dogs or arrange a face-to-face meeting to "settle" a fence argument.
Make the line less exciting
An opaque screen can block a sightline, and a second barrier can keep dogs from reaching the shared fence. Choose materials that are secure for your yard, and follow local property rules. A visual barrier will not stop every reaction to sound or smell, but it can remove one powerful cue. If the neighbor is willing, a simple conversation about alternating yard times may spare both dogs repeated rehearsals. Don't assume they will change their routine, make a plan that works on your side regardless.
Teach an easy recall or "inside" cue away from the fence. Reward often when your dog comes in during calm moments. Then practice at times the neighbor's dog is not outside. Calling once a full barking match is underway is much harder than interrupting the first glance toward the fence.
For training, stay far enough from the boundary that your dog can notice the other dog's presence and still eat, sniff, and return with you. Reward that calm notice and leave before the run begins. If the dogs have tried to bite through the fence, redirect onto people, or cannot be kept safely apart, work with a qualified, reward-based behavior professional. The goal is a peaceful yard, not a forced friendship across a barrier. For more on interrupting a rehearsed barking pattern, see the barking and reactivity guide, the fence must remain secure and the dogs safely apart
Sources: Best Friends Animal Society: How to Use Visual Barriers for Dog Reactivity. American Kennel Club: How to Stop Dogs From Fighting Through a Fence.