For creating the joint posterior distribution for multiple variables, are the associated Bayesian priors usually assume independent of each other?

For creating the joint posterior distribution for multiple variables, are the associated Bayesian priors usually assume independent of each other?

Suppose that we have data, D, and two parameters we want to learn about, theta_1, theta_2. We will usually put priors on theta_1, theta_2, then have the expression: ptheta_1, theta_2mid D propto pDmid theta_1, theta_2 ptheta_1 ptheta_2 I am wondering why most set-ups assume that the priors above are independent. What happens if we do not have it