Hi David,

I do not know of any results of the form you suggest.

There is a mathematical reason why these results are not too common, namely, the best known techniques to produce consistency results use either forcing axioms or inner model considerations. By a well-known absoluteness result of Shoenfield (see for example Section 13 of Kanamori's "The higher infinite"), all the models obtained this way satisfy the same $\Sigma^1_2$ statements. In particular, they satisfy the same arithmetic statements. This means that we cannot expect to arrive at models of $\lnot$Consist(ZFC) using these methods. But these are almost all the methods we have at our disposal!

Obtaining models with $\lnot$Consist(ZFC) or something similar, in addition to whatever interesting topological fact one is after, requires techniques for building (interesting) models that are not $\omega$-models, and would probably require that we leverage this strong ill-foundedness to our advantage. Until very recently there was no systematic approach to doing this, so the possibility, though interesting, was in essence intractable.

Recent work of Harvey Friedman suggests that this may change. At the moment, Friedman has used his techniques very effectively mostly to analyze certain combinatorial problems. His methods in a natural way require the construction of not-$\omega$-models (beginning with something like the negations of the combinatorial principle under study), and it is not too clear that one could do the same systematically with topological problems.

Friedman writes:

[...] it became clear that according to conventional wisdom, the
Incompleteness Phenomena was confined to questions of an
inherently set theoretic nature that was highly non
concrete, and out of touch with normal mathematical
activity.

[...] It was already clear to me at that time that despite the
great depth and beauty of the ongoing breakthroughs in set
theory regarding the continuum hypothesis and many other
tantalizing set theoretic problems, the long range impact
and significance of ongoing investigations in the
foundations of mathematics is going to depend greatly on
the extent to which the Incompleteness Phenomena touches
normal concrete mathematics. This perception was confirmed
in my first few years out of school at Stanford University
with further discussions with mathematics faculty,
including Paul J. Cohen.

[...] there were no candidates for Concrete Mathematical
Incompleteness from ZFC being offered. In fact, to this
day, no candidates for Concrete Mathematical Incompleteness
have arisen from the natural course of mathematics.

[...] The second rationale for pursuing Concrete Mathematical
Incompleteness preserves ZFC as the ambitious target. The
idea is that normal mathematical activity up to now
represents only an infinitesimal portion of eventual
mathematical activity. Even if current mathematical
activity does not give rise to Concrete Mathematical Incompleteness from ZFC, this is a very poor indication of
whether this will continue to be the case, particularly far
out into the future.

The tone, though cautions, is optimistic. The techniques Friedman has developed are presented in his book "Boolean relation theory", a draft of which can be found in Friedman's homepage. The quote above is from this book. Friedman's results typically require the construction of not-$\omega$-models, so studying his methods seems the most promising route towards realizing something like what you ask about.

Now, even in this case, his results do not typically land you at the level of $\lnot$Consist(ZFC), and it may require a couple of words to see their relevance.

A typical result has the form that certain combinatorial statement $P$ is equiconsistent with certain extension of ZFC by large cardinals. One direction is dome directly (even if the arguments are sophisticated): The combinatorics of the large cardinals allow us to obtain $P$.

However, $P$ tends to be of low complexity (say, it is arithmetic) so building from $\lnot P$ certain models where the consistency of some large cardinals fails requires that the models built are not $\omega$-models; it is here that several of Friedman's new techniques appear. In other words, we know in the standard model that $P$ holds, and it is in pathological models where the opposite is the case. To conclude, this indicates to me that even if not yet directly from current methods, these techniques seem the right direction to study if one hopes to eventually obtain a result of the kind as you suggest.