What is an example of a real world relationship where there is no clear independent or dependent variable?
This situation is actually more common than its opposite. It's quite rare in real-world research to have a clear idea of the direction of causality, and quite common to find strong correlations between variables where causation could run in either or both directions.
Here are a few examples just from my own field of development economics:
1. Corruption is strongly correlated to income inequality. Does corruption lead to inequality? Does higher inequality create more corruption?
2. GDP growth is strongly correlated with the empowerment of women. Does empowering women improve economic growth? Does an improving economy provide more opportunities for women?
3. Ethnic conflict is strongly correlated with poverty. Does greater ethnic conflict cause poverty? Does poverty exacerbate ethnic tensions?
In all these cases, it's also quite possible that both effects are true, or even that neither effect is true and a common cause of both drives the correlation. (For example, height is strongly correlated with weight in children. Does weight cause height or height cause weight? Well, neither, really; growth causes both height and weight.)
Here are some more examples from other fields as well:
1. Political science: Democracies are less likely to go to war with one another than authoritarian governments are. Does being democratic make a country less likely to go to war? Does being in a state of peace encourage the establishment of democratic institutions?
2. Sociology: Higher education is strongly correlated with lower rates of sexism. Does education make people less sexist? Or does being sexist discourage people from taking on higher levels of education?
3. Psychology: Higher socioeconomic status is correlated with lower rates of empathy. Does becoming rich make people less empathetic? Or are people with less empathy more likely to get rich?