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The main result is a wall crossing formula for central projections defined onsubmanifolds of a real projective space. Our formula gives the jump of thedegree of such a projection when the center of the projection varies. The factthat the degree depends on the projection is a new phenomenon, specific to realalgebraic geometry. We illustrate this phenomenon in many interestingsituations. The crucial assumption on the class of maps we consider is relativeorientability, a condition which allows us to define a $\Z$-valued degree mapin a coherent way. We end the article with several examples, e.g. the poleplacement map associated with a quotient, the Wronski map, and a new version ofthe real subspace problem.

A wall crossing formula for degrees of real central projections