I'm a bit confused working with Teichmuller space at the moment. Let's think of Teichmuller space as the space of holomorphic/conformal structures on a surface, up to diffeomorphisms isotopic to the identity. Consider the Teichmuller space of the 4-punctured sphere $T(S_{0,4})$ where I think of the punctures as being indistinguishable.
I can think of the four punctures as being four points in $S^2 = \mathbb{CP}^1$. Up to Mobius transformation, I can assume that two of the points are $0,\infty \in \mathbb{CP}^1$. Rotating about the origin in $\mathbb{C}$ then allows me to assume that one point lies in $\mathbb{R}^\times$ and the last lies in the upper half plane $\mathbb{H}$. I can impose some normalization, like asking that these last two points span a triangle of area $1$, to take care of positive-real scaling. Then the point in Teichmuller space is determined uniquely by this point in the upper half plane, i.e. $T(S_{0,4}) = \mathbb{H}$.
Or at least that's how it's supposed to go. But here's what I'm confused about: what happens when I take that fourth point down to the real line so that all four points lie on $\mathbb{RP}^1 \subset \mathbb{CP}^1$? If I'm normalizing to have this unit area condition, then things blow up when I try to line everything up like that. But of course puncturing e.g. $0,1,2,\infty$ in $\mathbb{CP}^1$ gives some complex structure on $S_{0,4}$. Am I missing something in my understanding of the Teichmuller space?
The area normalization is a bad idea.
A much better normalization is to require the first three points to go to $0$, $1$ and $\infty$ respectively. The position of the 4th point is then a conformal invariant, for the following simple reason: the identity is the unique conformal map of the Riemann sphere that fixes $0$, $1$ and $\infty$; and therefore for $w \ne z \in \mathbb C^* - \{0,1,\infty\}$ there is no conformal map taking $z$ to $w$.