We describe the long-term evolution of compact systems of terrestrial planets, using a set of simulations that match the statistical properties of the observed exoplanet distribution. The evolution is driven by tidal dissipation in the planetary interiors, but the systems evolve as a whole due to secular gravitational interactions. We find that, for Earth-like dissipation levels, planetary orbits can be circularized out to periods ~100 d, an order of magnitude larger than is possible for single planets. The resulting distribution of eccentricities is a qualitative match to that inferred from transit timing variations, with a minority of non-zero eccentricities maintained by particular secular configurations. The coupling of the tidal and secular processes enhance the inward migration of the innermost planets in these systems, and can drive them to short orbital periods. Resonant interactions of both the mean motion and secular variety are observed, although the interactions are not strong enough to drive systemic instability in most cases. However, we demonstrate that these systems can easily be driven unstable if coupled to giant planets on longer period orbits.