Various technological applications for body posture correction have been proposed in order to improve handwriting or facilitate its learning for children, under the common-sense assumption that a better posture promotes better handwriting. However, very little research investigates the correlation between body posture quality and handwriting quality. Moreover, investigating this correlation typically necessitates the expertise of human observers, leading to high costs, slow progress, and potential subjectivity issues. Consequently, this method may not be suitable for educational environments that require prompt feedback and interventions. In this paper, we present a fully-automated pipeline for the real-time assessment of body posture quality, which builds upon validated scales from ergonomics, which relies on RGB-D data to compute the REBA/RULA body posture scores. Together with a state-of-the-art tool for the automated, real-time assessment of handwriting quality, we applied our pipeline in an experiment at school involving 31 children, to quantitatively and objectively investigate (i) the correlation between body posture quality and handwriting quality, as well as (ii) the impact that interventions aimed at improving the children's body posture have on their handwriting quality.