Egg color is determined by the amount of antimicrobial secretion on the eggshell.
Male birds often go to great lengths to attract the attention of female birds. Some bird-of-paradise species exhibit eccentric courtship displays to woo the females. And bowerbirds build elaborate structures with twigs and leaves to convince potential partners. Other species keep it simple and only show off their bright colors. No matter what strategy evolved, in each case the males are advertising their quality. For instance, the tail of a peacock indicates that this male manages to escape predators and stay parasite-free despite dragging such a long and colorful structure behind him. Surely, this bird has very good genes that he can pass on to the next generation. Females choose their partner based on the quality of such traits and thereby influence the evolution of male morphology and behavior. This idea was put forward in 1871 by Charles Darwin and is now generally known as sexual selection.
You might have noticed that the examples above are very male-biased. The male birds dance, strut their feathers and sing, while the female birds silently judge and pick a winner. But how does the male know that he is being picked by a good quality female? In other words, can females also advertise their quality?

The colorful feathers of the Mandarin duck advertise the quality of this male. © Alexandra Sora | Wikimedia Commons
Egg Color
This question is particularly relevant in bird species where males provide a lot of parental care. If you are going to invest time and resources in raising young, you need to be sure that your partner is worth it. One way in which males could judge the quality of the female is by inspecting the color of the eggs. For example, the blue-green color of eggs is due to biliverdin, a strong anti-oxidant. Females that lay intense blue-green eggs might thus be advertising their anti-oxidant capacity.
A recent study in the Journal of Avian Biology applied this reasoning to Hoopoes (Upupa epops). Females of this species stain their eggs with a secretion from the uropygial gland (also known as the preen gland, located at the base of the tail). This secretion contains symbiotic bacteria that protect the eggs from pathogens. By applying this antimicrobial substance, the eggs also change color: from bluish to greenish-grey. Hence, the male Hoopoe could judge the quality of the female – in terms of protective bacteria – by looking at the color of the eggshells and adjust his parental care accordingly.

The eggs of a Hoopoe © Didier Descouens | Wikimedia Commons
Saturation Levels
This leads to a clear and testable hypothesis: male Hoopoes should bring more food to nests with greenish-grey eggs. Silvia Díaz Lora and her colleagues tested this prediction in Spain by observing 39 Hoopoe nests. First, they established the relationship between egg color and symbiotic bacteria. The color of the eggs was standardized by measuring their saturation (rA). A high saturation points to a brightly colored egg, while a low saturation indicates a grey or white egg. As expected, the saturation of the eggs was negatively related to bacterial density in the uropygial secretion. In other words, the lower the saturation of the eggs (i.e. grey eggs), the more symbiotic bacteria on the eggshell. Next, the researchers correlated the saturation of the eggs with the provisioning effort of the male. And in accordance with their prediction, males brought more food to nests with grey eggs (i.e. low saturation and thus more symbiotic bacteria).
These findings support the idea that females can advertise their quality through the color of the eggs. However, we cannot be completely sure that male Hoopoes are actively inspecting the egg color. Perhaps the relationship is determined by a third factor that is related to female quality. It is possible that males focus on another trait that covaries with number of symbiotic bacteria deposited on the eggshells. Experimental manipulation is needed to check whether the correlation in this study is an actual causation.

The relationship between male provisioning rate and eggshell saturation indicates that males brought more food to nests with lowly saturated eggs (i.e. white or grey eggs). From: Diaz Lora et al. (2020) Journal of Avian Biology.
References
Díaz Lora, S., Pérez‐Contreras, T., Azcárate‐García, M., Martínez Bueno, M., Soler, J. J., & Martín‐Vivaldi, M. (2020). Hoopoe Upupa epops male feeding effort is related to female cosmetic egg colouration. Journal of Avian Biology, 51(8).
Featured Image: A Hoopoe in Hungary © Andy Morffew | Wikimedia Commons