Imagine a world in which the gambler's fallacy is fundamentally true. Functionally, lets suppose there's a magical force that tracks a thinking being's expectation of any particular outcome, and then mysteriously increases the likelihood of said outcome the more often it had physically plausible opportunity to occur and did not[1].

A more natural way to implement this might be to avoid the thinking part and simply say that in this world, there is no sampling-with-replacement, there is only sampling-without-replacement. All 'independent' events are dependent, because 'randomness' is actually pregenerated shuffled lists which get used up one by one, and earlier events now change your best prediction of future events due to the underlying dependence on the hidden list of randomness. So before you flip a fair coin 100 times, what happened was that 50 heads and 50 tails were generated, and shuffled; if you flip and get 10 heads in a row, you now expect there to be 40 heads and 50 tails left and the next flip to be heads with only 40/(40+50) probability and so "tails is due!" This continues until you've flipped 100 times, at which point a new shuffled list will govern any future flips, and your expectation resets to 50-50. This gives us the classical gambler's fallacy, which Wikipedia defines as:

The gambler's fallacy, also known as the Monte Carlo fallacy or the fallacy of the maturity of chances, is the belief that, if an event (whose occurrences are independent and identically distributed) has occurred less frequently than expected, it is more likely to happen again in the future (or vice versa). The fallacy is commonly associated with gambling, where it may be believed, for example, that the next dice roll is more than usually likely to be six because there have recently been fewer than the expected number of sixes.

That is, it's not simply an expectation of some sort, it's the specific expectation that the next random event is going to regress back to the mean - if you've had 'too many' heads, then the next flip 'should' be tails. This avoids the issues with expectations - whose expectations, when? - and replaces it with something you could actually write down a computable version of*: you simply figure out how to associate 'random' events with an appropriate shuffled PRNG, and now you have a well-defined alternative physics where the gambler's fallacy is true. (There would still be cases where you'd act as if it's false and you were in our world, but these would be due to more complex situations, like ones where you were unsure what the bias of the coin was and your posterior over the bias counterbalanced the changing probability, or ones where you were unsure of the period or length of the hidden randomness and so your posterior over all of the changepoints offset your posterior on the list contents - if the list had just changed, then you reset to the gambler's fallacy.)

So, in the Gambler's Verity world, you can do things like manufacture 'lucky' dice by rolling many dice, and keeping the ones which have 'used up' the most unlucky outcomes. (I hear D&D players do this as a joke, but in this world, it might actually work.) You would no longer be able to flip fair coins easily because whoever provided the coin could've provided one pre-flipped to yield the desired outcome; you would have to use alternative methods, like both parties flipping their own coin simultaneously and using a randomness extractor on the pair of results. You also are able to more profitably exploit merely 'fair' opportunities because the odds will change and have option value (eg. Problem 14). Depending on the granularity of the hidden variables and what micro or macro-states they hold over, you could imagine investing being very different: instead of efficient markets driven by random walks at each instant, you'd get to efficiency by instead pricing in risk premium - 'good' companies have their stock prices systematically lowered because they are due for a run of bad luck, while 'bad' companies' in contrast enjoy a premium because they may be about to embark on a bull run. Forecasting & analytics become much more powerful & valuable, and it becomes worth tracking everything possible, because you may be able to identify time series and move in and out to manage the risk; statisticians will warn you about your lucky and unlucky days, and you will avoid going out on inauspicious days where you might be run over by a horseless carriage. There are probably weird consequences in thermodynamics & physics from these hidden variables too, but I'm not sure what. (Is this a local hidden-variable theory? Superdeterminism? Can you violate thermodynamics by Maxwell's demon here to gain free energy, or does the tracking of history still wind up erasing the gains? etc)

* and, AFAIK, this is actually something that is done; aside from topics in numerical analysis or physics where you use "quasi-random" number generators or other biased kinds of randomness to ensure a 'more even' coverage and gain efficiency, IIRC, games will often implement randomness in a sampling-without-replacement way, to cater to players' prejudices and ensure more fun. It's not fun to get a long run of 'bad' random outcomes, if there is nothing which counterbalances that; card games rely heavily on this as a mechanic, where if you get a lot of 'bad' draws from the deck, you can take consolation in the fact that the remaining deck is enriched for 'good' cards, automatically tempering the extremes and adding a layer of strategy. This is also often done by bending the probabilities, which implies a Gambler's Verity world: if a player is low on health or doing badly, they'll find that they start beating impossible odds like they're Han Solo, and so lots of bad outcomes will in fact imply that they are then 'due' for good outcomes.

The Gambler's Fallacy applies in both directions: if an event has happened more frequently in the past than expected, then the Fallacy states that it is less likely to occur in future as well. So for example, rolling a 6-sided die three times and getting two sixes in such a world should also decrease the probability of getting another six on the next roll by some unspecified amount.

That is, it's a world in which steps in every random walk are biased toward the mean.

However, that does run into some difficulties. Suppose that person A is flipping coins and keeping track of the numbers of heads and tails. The count is 89 tails to 111 heads so far. Person B comes in watches for 100 more flips. They see 56 more tails and 44 heads, so that A's count is now at 145 tails to 155 heads. Gambler's Verity applied to A means that tails should still be more likely. Gambler's Verity applied to B means that heads should be more likely. Which effect is stronger?

Now consider person C who isn't told the outcomes of each flip, just whether the flip moved the counts more toward equal or further away. Gambler's Verity for those who see each flip means that "toward equal" flips are more common than "more unequal" flips. But applied to C's observations, Gambler's Verity acts to cancel out any bias even more rapidly than independent chance would. So if you're aware of Gambler's Verity and try to study it, then it cancels itself out!

I'm not sure how this would work, especially in conflicting-expectation cases (one person watches a die roll 20 times, another person only saw the last 5, they have DIFFERENT expectations in this non-independent-trials world.  What actually happens with what frequency?  Depending on specifics, presumably casinos hire people to differentially watch games and preferentially shuffle or change the dice.  

Other good effects: nobody expects to get cancer, so I guess it doesn't happen?  

But really, it's just confused in conception - this fallacy is based on a reference class - events are no longer independent, but it's unclear in what way they're entangled.  If I roll 12 different dice, does it apply?  If I roll the same die 12 times, but only observe the first 2 and the last, do the 8 unobserved rolls affect the expectation?