MAX 1 in 2 SAT Algorithm
A monotone 1-in-2 clause demands that the two variables have different values. Thus, you can model the problem as a graph problem, with one vertex per variable which is to be colored black or white, and an edge for a clause indicating the colors need to be different. Thus, the question is to make the graph bipartite by deleting a minimum number of edges. This is the MaxCut or Edge Bipartization problem. It is NP-hard.

For Edge Bipartization, there is an algorithm that runs fast when few edges need to be deleted. I wrote an implementation for a slightly more general problem described here (source code).