What kind of solution is there to this? Only trust studies which have been successfully reproduced several times? Raise the bar for an acceptable p value so it becomes less practical to p-hack?
This really should be the standard for any meaningful science.
Submiting a large number of individual registrations for a single set of data is just not the norm and so it would be suspicious to begin with. (Okay, I will admit that on the order of 2-3 registrations for very different categories of study may not be as strange; but certainly 20 would signal something very odd).
That said, you can do something called multiple comparisons corrections, where you take into account the fact that you're doing multiple tests in the same go. In this case you'll end up being more conservative, and it's not the false positive rate you'll be controlling - instead you'll control the family-wise error rate, which basically has to do with whether _any_ of your positive results are false positives.
It's much much harder to lie with effect sizes. For instance, in genetics there are entire careers built on an association of certain genetic polymorphism with a particular phenotype. Even if the p-value can be reproduced with huge sample sizes, the effect size of the association is usually tiny.
In other words, if you carry the risk allele the odds of developing a disease are usually only slightly bigger than if you didn't. But this is often not told to funders! Focusing on effect sizes would drive research effort to things that actually make a difference.
You can do effect sizes with something as simple as a confidence interval, or with a fully fledged Bayesian model that tries to estimate the posterior distribution of some parameter.
While impractical in many cases, I think the gold standard is to approach a problem from as many angles as possible, and relate it to disparate sources of knowledge, including theory.
For example, the accepted values of the physical constants are established by munging the results from a wide variety of different kinds of experiments, guided by theory. They don't just repeat the same experiment over and over. These constants are robust enough that you can hang your hat on them for virtually any practical use.