Complexity results about locally bipartite graphs
Locally bipartite graphs obviously contain the locally independent (= triangle-free) graphs. According to graphclasses.org, most of the standard graph problems are already NP-complete for triangle-free graphs, and therefore also NP-complete for locally bipartite graphs. The two exceptions are clique (which is obviously polynomial for locally bipartite graphs because the maximum clique is a triangle) and clique cover, which is polynomial for triangle-free graphs but might be harder for locally bipartite graphs.