Suppose for simplicity's sak that $A$ and $B$ are diagonalizable over $\mathbb{R}$ and are non-singular.

Let $V_a$ be the $a$-eigenspace of $A$. Then by anti-commutativity, we find $AV_a\subseteq V_{-a}$ etc. As $A$ and $B^2$ commute then there is an eigenvector $v\in V_a$ with $B^2v=b^2 v$ for some $v$. If we let $w=bv+Bv$ then $Bw=bw$ so $b$ is real (assuming $B$ has real eigenvectors). On the space $W$ spanned by $v$ and $w=b^{-1}Bv$ the linear transformation $A+B$ has matrix $$\left(\begin{array}{rr} a&b\\\\ b&-a\\\\ \end{array}\right)$$ which has an eigenvectors with eigenvalues $\pm\sqrt{a^2+b^2}$.

Suppose for simplicity's sak that $A$ and $B$ are diagonalizable over $\mathbb{R}$ and are non-singular.

Let $V_a$ be the $a$-eigenspace of $A$. Then by anti-commutativity, we find $BV_a\subseteq V_{-a}$ etc. As $A$ and $B^2$ commute then there is an eigenvector $v\in V_a$ with $B^2v=b^2 v$ for some $v$. If we let $w=bv+Bv$ then $Bw=bw$ so $b$ is real (assuming $B$ has real eigenvectors). On the space $W$ spanned by $v$ and $w=b^{-1}Bv$ the linear transformation $A+B$ has matrix $$\left(\begin{array}{rr} a&b\\\\ b&-a\\\\ \end{array}\right)$$ which has an eigenvectors with eigenvalues $\pm\sqrt{a^2+b^2}$.