Is there such an example? Or it is known that a pseudo algebraically closed field which is a finite extension of a formally real field is algebraically closed?
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As far as I can tell, you can take your formally real field to be the field $\mathbb{Q}^{tr}$ of totally real algebraic numbers (see this paper for a description of its Galois group). Then (according to Wikipedia), adjoining a square root of $1$ gives you a pseudo algebraically closed field. 

