Rational functions with trivial Weil symbols at every point

Let $f, g$ be a pair of nonzero rational functions in $mathbb{C}(t).$ For $lambdain mathbb{C}$ let $a$ be multiplicity of $g(t)$ at $lambda$ and $b$ – multiplicity of $f(t)$ at $lambda.$ Weil symbol of $f$ and $g$ at $lambda$ is defined by the following formula:
$$
(f,g)_{lambda}=(-1)^{ab}frac{f^a}{g^b}(lambda).
$$

Question: For which pairs of rational functions $f, g$ Weil symbol $
(f,g)_{lambda}$ equals to $pm1$ at every point $lambdain mathbb{C}?$

It is easy to find some pairs of such functions. For every $r(t)in mathbb{C}(t)$ we can take $f=r(t)^a(1-r(t))^b$ and $g=r(t)^c(1-r(t))^d.$

This problem can be generalized to arbitrary Riemann surfaces, but it is probably very hard, because nontrivial examples of such pairs of functions come from $A-$polynomials of some knots.