Suppose \(X\) is an algebraic curve over the real field of degree \(d\) embedded into some projective space. One can ask whether there exists a real hyperplane meeting \(X\) in real points only. More generally, given any divisor \(D\) on \(X\) defined over \(\mathbb{R}\), one may ask whether the linear series \(|D|\) contains an effective divisor with totally real support. Note that the linear series \(|D|\) contains an effective divisor with totally real support if the degree of the divisors is sufficiently large. (See [V. A. Krasnov, Math. Notes 32, 661–666 (1982; Zbl 0515.14028); translation from Mat. Zametki 32, 365–374 (1982); C. Scheiderer, Trans. Am. Math. Soc. 352, No. 3, 1039–1069 (2000; Zbl 0941.14024)].) Moreover, it is also known that for a given genus and number of real connected components, any linear series of sufficiently large degrees contains a totally real effective divisor.
Even though the theoretical answer to the problem is positive, one may curious about explicit bounds for the degree of the divisors (such that the linear series \(|D|\) contains an effective divisor with totally-real supports).
By the way, the problem can be translated into a particular type of parametrized real root counting problem and they introduce an algorithm for solving the parametric systems. Using the algorithms described in this paper, they investigate a number of examples, which we can compare to the best-known bounds for the required degree.
Here is the summary of their computational results:

1.

There exist canonical curves \(X\) in \(\mathbb{P}^3\) with one or two ovals which do not allow simple totally real hyperplane sections (Example 3.1).

2.

There exists a curve \(X\) in \(\mathbb{P}^3\) of genus two and degree five having one oval which does not allow a simple totally real hyperplane section (Example 3.3).

3.

There are infinitely many plane quartics X with many ovals possessing a (complete) linear series of degree four which does not contain a totally real divisor (Example 4.2).

4.

For every \(d \ge 3\) and every number \(1 \le s\le g+1\) with \(g=\frac{(d-1)(d-2)}{2}\), there exists a plane curve \(X\) of degree \(d\), genus \(g\) and having \(s\) branches such that the linear series of lines \(|L|\) is totally real (Theorem 4.3).