Assume A and B are both positive. Let K be the ratio (1 + 1/A)/(1 + 1/B). Rewrite K as [B*(A+1)]/[A*(B+1)]. A=B, K=1 is a trivial solution.
A and A+1 are coprime, so are B and B+1. Then for K to be an integer with A!=B, a must divide B and B+1 must divide A+1. These imply B>A and A>B simultaneously. No value of A and B can satisfy that requirement so there are no solutions with A!=B and both A and B both positive.
Assume A and B are both negative. Let X=1-A and Y=1-B. X and y will both be positive. Then K = [Y*(X-1)]/[X*(Y-1)]. This leads to the same conclusion as the previous case, the only solutions have A=B and K=1.
Assume A is negative and B is positive. Then 0 < (1 + 1/A) < (1 + 1/B), which means K can never be an integer. No solutions in this case.
This leaves A is positive and B is negative. Charlie's earlier work found five nontrivial solutions. Now to prove those are the only ones....