Godel's first theorem says no formal system with recursively enumerable axioms and which is powerful enough to express elementary arithmetic can be both consistent and complete. Most people adopt the response – alas, that means no formal systems we can devise can ever be complete (except for systems too weak to express arithmetic, which are not very useful). But, paraconsistency and dialetheism give us another response: yes, we can have complete formal systems with recursively enumerable axioms and which are powerful enough to express elementary arithmetic. Okay, they'll be inconsistent, but inconsistency is not as bad as you thought it was. We can contain the inconsistency to some small area in which it isn't going to bother you, you'll scarcely notice that it is there. We can hide the contradictions in the closet and go on our merry way not thinking about their existence except on rare occasions.

If mathematics is inconsistent, does that make mathematics objectively unreal? Only if we insist objective reality must be consistent. If objective reality contains dialetheias, then inconsistent mathematics can be objectively real.