Let $X$ be the subspace of $3 \times 2$ complex matrices with full rank, where we identify matrices differing from one another by a scalar. There are some natural maps from $X$ into $\mathbb{C}\!\operatorname{P}^2$, namely
$$\begin{pmatrix} a & b \\ c & d \\ e & f \\ \end{pmatrix} \mapsto [ a:c:e ] \qquad \text{and} \qquad \begin{pmatrix} a & b \\ c & d \\ e & f \\ \end{pmatrix} \mapsto [ \overline{cf-de}:\overline{be-af}:\overline{ad-bc} ].$$
The first map is clearly given by evaluating at $e_1$ and passing to projective space, while the second is given by taking the column span and then computing the normal vector to that plane via cross product (this is of course the identification of the Grassmannians of 2-planes in $\mathbb{C}^3$ with $\mathbb{C}\!\operatorname{P}^2$ via orthogonal complement).
Neither of these maps should admit global sections, but I’m trying my hand at the next best thing. You might be tempted, for the first map, to define a section as follows:
$$[ u:v:w ] \mapsto \begin{pmatrix} u & 1 \\ v & 0 \\ w & 0 \\ \end{pmatrix}.$$
This would be defined away from $[1:0:0]$, but of course for this map to actually make sense each entry would need to be a homogeneous polynomial (of the same degree) in the $u$, $v$, $w$, and their conjugates. As an actual example, when we consider the first map but think of $2\times2$ matrices instead we can make a section
$$[u:v] \mapsto \begin{pmatrix} u\overline{v} & -\overline{v}v \\ v\overline{v} & \overline{u}v \\ \end{pmatrix}$$
away from $[1:0]$. Thinking of the domain as $S^2\backslash \infty = \mathbb{C}$, we’ve written
$$r e^{\theta i} \mapsto \begin{pmatrix} \tfrac{re^{\theta i}}{\sqrt{1+r^2}} & -\tfrac{1}{\sqrt{1+r^2}} \\ \tfrac{1}{\sqrt{1+r^2}} & \tfrac{re^{-\theta i}}{\sqrt{1+r^2}} \\ \end{pmatrix} \in \operatorname{PSL}_2(\mathbb{C}).$$
In general, I want to study $n \times 2$ matrices and maps $X \to \mathbb{C}\!\operatorname{P}^n$ and $X \to \operatorname{Gr}(2,n+1)$. Finding sections when $n>2$ might be hopeless simply because of some basic linear algebra fact, but I’d appreciate any input.