Are there any fundamental obstacles to using a nuclear explosion as a trigger for antimatter generation and thus considerably increasing the yield?
Unfortunately, it wouldn't increase the yield.
When matter and antimatter annihilate each other, yes, they do liberate a huge amount of energy. But, you don't get energy for free - no matter how it's produced, creating antimatter will take up exactly as much energy as annihilating it will release.
So, even if it were plausible that a fission or fusion bomb could be rigged to produce antimatter, the antimatter annihilation would only give you back energy that it borrowed from the fission/fusion reaction in the first place.
Are there any fundamental obstacles to using a nuclear explosion as a trigger for antimatter generation and thus considerably increasing the yield
What you refer to is the yield limit (I believe it was first pointed out by Ulam) to a fusion device, caused by the explosion disrupting the conditions that allow fusion to take place. By carefully constructing the device so it explodes in "stages", the yield figure of around 7-8 megatons per ton can be maintained in the tens of tons range.
Beyond that, the explosion is no longer controllable and any additional fusion material (typically lithium deuteride) will only add a negligible amount to the bang, in the form of a "fizzle".
So, converting matter to anti-matter, which will explode no matter what - pardon the pun - however widely dispersed, would appear as a practical shortcut around the yield limit, even if the laws of conservation of energy mandate that the total maximum theoretical yield will be the same. Antimatter conversion would simply allow coming closer to that yield.
This is somewhat akin to the technique of injecting nebulized water into gas engines. While the total chemical energy available can't change (it actually decreases slightly), the deflagration becomes smoother, more thorough and better suited to supplying expansion work; the result is therefore an increase in power yield, through a reduction of wasted energy (and fuel).
...or rather, yes.
The fundamental obstacle lies in the conversion itself - there is no known process that can lead to the creation of matter/antimatter particle pairs from a fusion event with any efficiency.
Currently, antiprotons are produced by shooting very high energy protons (to the tune of 26+ GeV) against suitable targets, and the antimatter yield is about four antiprotons every million collisions. So we currently need to expend around 25,000 Teraelectronvolts to create four particles with a rest mass of around four GeV.
If the whole energy of the Tsar Bomba at full design spec (100 Mt) were to be converted into antimatter with such yields, we could squeeze a yield increase of maybe 1-2 Mt. It is entirely probable that to achieve this, though, the explosive power of the Bomba would be greatly diminished, so that the operation would turn out as a net loss.
Yes, there's a huge fundamental stumbling block.
Conservation of energy. The energy to create the antimatter has to come from the nuclear fission and/or fusion of the "trigger", so the net, final yield can't exceed that of the nuclear material in the original bomb.
Now, John W. Campbell once wrote a series of books in which, among other completely implausible things (even for the 1950s), he had a ray that converted 50% of matter to antimatter. And then, somehow, rather than erasing Chicago the first time it was tested, his character managed to turn it into an energy generator and propulsion system. This amounted to annihilating matter by converting it to its equivalent energy, but that's a whole different thing from using the (minuscule) quantity of matter/antimatter pairs produced in a nuclear explosion to boost the yield (they're already part of it).