Finsler $N$-spinors with real components

We study the mathematical objects called Finsler $N$-spinors over the field $\mathbb{R}$ and construct the general algebraic theory of these objects. We show that the Finsler $N$-spinors over the field $\mathbb{R}$ generate two families of $N(N{+}1)/2$- and $N(N{-}1)/2$-dimensional flat pseudo-Finsler spaces. We generalize the epimorphism $SL(2,\mathbb{R})\to O^\uparrow_+(1,2)$ to the case of the group $SL(N,\mathbb{R})$. We consider the examples of Finsler $N$-spinors over the field $\mathbb{R}$ for $N=2,3$ in detail.