During evaporative condition in a closed basin, there is a high correlation between d13C and d18O of bulk carbonate. What factors that influence isotopic composition of d13C during evaporation, so that it covary with d18O.
The 13C enrichments is inversely related to ΣCO2 and as evaporation proceeds an increase in salinity occurs with a concurrent decrease in Ph and ΣCO2 content. This suggests that evaporation effect related to the loss of CO2 from the brine causes the 13C enrichment (Stiller et al., 1985).
As the 18O enrichment is also dependant on the evaporation effect the high covariance can be explained in terms of evaporation effect.
Stiller et al. (1985) is a good classical example for an evaporitic system.
Both, 18O and 13C of the precipitated CaCO3 depend on the isotope composition of the dissolved carbonate group. Degassing of CO2 from supersaturated aqueous solution leads to the enrichment of 13C in residual DIC. Therefore, since both the distribution of 13C and 18O of the individual dissolved inorganic carbon species depend on pH (e.g., Beck et al., 2005) it will also vary with biological activity, salt contents (e.g., ion pairs-look on the original experimental literature from the 60s and 70s) and most likely precipitation rate. Biological activity may, therefore, also impact the signal via pH or kinetic effetcs as well as later recrystallization reactions.
Michael's answer is very helpful. I also note that the d13-CO3 signature of carbonates will also depend on the d13C composition of the original CO2 source. Terrestrial evaporites in volcanically active areas have more positive d13CO2 signatures (see work on Atacama Desert). Mantle degassing can also influence the initial d13CO2 signature in tectonically active areas. In such settings, covariance in d18O and d13CO2 is still expected during brine evolution but the resultant d13-CO3 signature will vary according to the CO2 source.