Consider the real Möbius bundle over $S^1,$ defined as follows. The bundle is trivial over $U_1 = S^1 \setminus \{i\}$ and $U_2 = S^1 \setminus \{-i\}$ and the transition function $T_{12}$ defined on $U_1 \cap U_2$ is given by $T_{12}(z) = 1$ when $\textbf{Re}\ z > 1$ and $T_{12}(z) = -1$ when $\textbf{Re}\ z < 1.$
Sections of this bundle can be considered as pairs of real valued functions $f_1, f_2$ defined on $U_1, U_2$ respectively such that $f_1(z) = f_2(z)$ when $\textbf{Re}\ z > 1,$ and $f_1(z) = -f_2(z)$ when $\textbf{Re}\ z < 1.$
Can we express the sections naturally as real valued functions defined on all of $S^1$satisfying some further condition? My guess would be that sections are functions satisfying $f(-z) = -f(z)$ or perhaps just $f(-i) = -f(i).$ More generally, can sections of a (real or complex) bundle always be expressed as globally defined real/complex valued functions satisfying some further conditions?