Suppose we do rejection sampling. Then the universe restarts at 2000 every time a datapoint is violated. The subjective experience of entities inside such a simulation would be best described as random death with occasional survival if some arbitrary criteria are met. Writing (long-lasting) believable false reports will alter the criteria to include those reports somehow being written.
MCMC algorithm is more interesting. It introduces a strange contortion of time; we generate an initially random space-time with the known datapoints clamped, and then re-sample the unknown bits for a long time to get a good posterior distribution over the possibilities. Real time is a sometimes-nonsensical meandering through possibility-space. Time as experienced "in the simulation" is relatively normal, but it is interesting/boring to argue about whether entities living in such a simulation have experiences in a meaningful sense. Causality doesn't work at all as it should, but it will appear to work roughly as it should most of the time. Events will conspire to bring about a random assortment of facts which are the clamped values, but the better-quality samples will make the conspiracies look like true chance.
Metropolis-hastings is an even more advanced technique, but I don't think there is anything special about the subjective experience if the simulated entities in MH as compared to basic MCMC.
While I've been wrestling with the inspiration needed to turn my fanfiction into actual fiction rather than just an author's tract... I have had an unrelated but fun thought, which I'm throwing into the breeze here for any improvements that can be suggested.
If our universe is more likely to be a simulation, a reconstruction of the past by our descendants than the base level of reality; then that reconstruction is likely to be imperfect, based mainly on surviving records (and memories of anyone whose brain survives intact long enough for upload-style scanning).
Therefore, if somebody precommits to only leave behind records which correspond to particular events... then it seems plausible (to within the bounds of 'the brain is a quantum computer' levels of plausibility) that those events become more likely to be experienced. For example, if a protagonist were to precommit to mentioning that during their walk, a bird landed right in their hand to eat a bread crust, whether or not such an event actually happened; then the probability that they will then experience a bird landing in their hand increases.
There are, of course, extreme limits on what can be accomplished with such trickery, even in theory. Violating the known laws of physics is right out, as are events dramatic enough to leave behind more traces than their own memories and journals. It also seems highly recommended for someone who wishes to try this to precommit to never leaving a trace that they are going to try it, as that would leave a record the future simulationists could use to discount their less-probable reports.
So - what additional thoughts could be added to the above to make it more plausible, at least to those who've heard of simulationism in the first place?
(And since it seems more likely than not that someone will ask: No I've never tried using such simulationist sympathetic magic myself, and since I still question the basic assumptions behind such mass-simulation in the first place, I have no intention of trying it in the future, either.)