We consider a general causal relativistic theory of divergence type in the framework of rational extended thermodynamics (RET) for a compressible, possibly dense, gas. We require that the system converges in the Maxwellian iteration's first step to the parabolic Eckart equations. This requirement implies a constraint between the two coefficients present in the triple tensor evaluated at equilibrium. Moreover, the production tensor is determined for prescript thermal and caloric state equations and given heat conductivity, shear and bulk viscosities. In the second part, we prove that if the original hyperbolic system satisfies the universal principles of RET, as can be put in the symmetric form using the main field, it always satisfies the previous compatibility condition. Therefore, any causal system of divergence type that satisfies the entropy principle with a convex entropy converges to the Eckart system in the Maxwellian iteration also when we have no information at the mesoscopic scale from the kinetic theory. The obtained results are tested on the RET theories of rarefied monatomic and polyatomic gases. This article is part of the theme issue 'Foundational issues, analysis and geometry in continuum mechanics'.
Keywords: 35L40; 76N10; 76N15; 82C35.