I'm wondering whether the order of events can lead to different Bayesian update.

For example, consider a coin-tossing problem with unknown $p$, the probability of Head. Initially, $p$ is known to follow some beta distribution: $$p\sim Beta(a_0,b_0).$$ Suppose that we have a sequence of observations that do not have to be an outcome of coin-tossing. For example, the first observation is "$\mathbb E[p]>\frac{1}{2}$" while the second observation is "Head".

If I want to update $p$ using Baye's rule, it will be a lot easier if I can process the second event first and then the first event later as Beta is a conjugate prior of binomial experiments.

However, if I have to update $p$ in the order of the events (first observation first, and then the second one later), the process requires a bit more of computation.

So, my question is that does the order of events matter in Bayesian updating? If not, what can be a theoretical background that justifies it?