Fiber-reinforced ceramic-matrix composites are advanced materials resistant to high temperatures, with application to aerospace engineering. Their analysis depends on the detection of embedded fibers, with semi-supervised techniques usually employed to separate fibers within the fiber beds. Here we present an open computational pipeline to detect fibers in ex-situ X-ray computed tomography fiber beds. To separate the fibers in these samples, we tested four different architectures of fully convolutional neural networks. When comparing our neural network approach to a semi-supervised one, we obtained Dice and Matthews coefficients greater than $92.28 \pm 9.65\%$, reaching up to $98.42 \pm 0.03 \%$, showing that the network results are close to the human-supervised ones in these fiber beds, in some cases separating fibers that human-curated algorithms could not find. The software we generated in this project is open source, released under a permissive license, and can be freely adapted and re-used in other domains. All data and instructions on how to download and use it are also available.