To answer the question "Why does Marching Tetrahedrons algo has ambiguities?" it is required to understand why do the ambiguities arise in the first place in Marching Cubes.
Ambiguities may occur when there are two diagonally opposite "positive" vertices and two diagonally opposite "negative" vertices in a cube. It took me some time to wrap my mind around it, but the problem with ambiguities is that they theoretically allow to create isosurface patches for adjacent cubes that are incompatible with one another. That's the obvious part. The interesting part is that two adjacent isosurface patches from two ambiguous configurations are incompatible if (and only if) one of them separates "negative" vertices, and the other one separates "positive" verticies.
Here is a relevant quote from Rephael Wenger's great book "Isosurfaces Geometry, Topology & Algorithms" (can't post more then 2 links, so I've merged all relevant images from the book into a single one):
The border of a cube’s three-dimensional isosurface patch defines an
isocontour on each of the cube’s square facets. If some configuration’s
isosurface patch separates the negative vertices on the facet while an
adjacent configuration’s isosurface patch separates the positive ones,
then the isosurface edges on the common facet will not align. The
isosurface patches in Figure 2.16 do not separate the positive
vertices on any facet. Moreover, the derived isosurface surface
patches in any rotation or reflection of the configurations also do not
separate positive vertices on any facet. Thus the isosurface patches
in any two adjacent cubes are properly aligned on their boundaries. An
equally valid, but combinatorially distinct, isosurface table could be
generated by using isosurface patches that do not separate the
negative vertices on any square facet.
What this means is that if all used ambiguous configurations follow the same pattern (i.e. always separate "negative" vertices), then it is impossible to produce a topologically incorrect surface. And problems will arise if you use configurations "from both worlds" for a single isosurface.
The surface that was constructed using the same ambiguity resolution pattern still may contain unwanted errors like this (taken from "Efficient implementation of Marching Cubes’ cases with topological guarantees" article by Thomas Lewiner Helio Lopes, Antonio Wilson Vieira and Geovan Tavares), but it will be, as you said, watertight.
To achieve this, you will need to use the look-up table based on the 22 unique configurations (not the standard 14 or 15) shown in the Figure 2.16.
Now, back to the original question at last - why does Marching Tetrahedrons does not suffer from ambiguities? For the same reason there will be no ambiguities in the Marching Cubes, if done as described above - because you arbitrarily chose to use one of the two available variants of ambiguous configuration resolution. In Marching Cubes it is not obvious at all (at least for me, had to do a lot of digging) that this is even an option, but in Marching Tetrahedrons it is done for you by the algorithm itself. Here is another quote from Rephael Wenger's book:
The regular grid cubes have ambiguous configurations while the
tetrahedral decomposition does not. What happened to the ambiguous
configurations? These configurations are resolved by the choice of
triangulation. For instance, in Figure 2.31, the first triangulation
gives an isosurface patch with two components corresponding to 2B-II
in Figure 2.22 while the second gives an isosurface patch with one
component corresponding to 2B-I.
Note how cubes are sliced into tetrahedrons in two different ways in Figure 2.31. The choice of this slicing or the other is the secret sauce that resolves the ambiguities.
One may ask himself - if the ambiguity problem can be resolved just by using the same pattern for all cubes then why are there so much books and papers about more complicated solutions? Why do I need Asymptotic Decider and all that stuff? As far as I can tell, it all comes down to what you need to achieve. If topological correctness (as in, no holes) is enough for you, then you do not need all the advanced stuff. If you want to resolve problems like those shown in "Efficient implementation of Marching Cubes" article shown above, then you need to dive deeper.
I highly recommend reading the relevant chapters of Rephael Wenger's book "Isosurfaces Geometry, Topology & Algorithms" to better understand the nature of these algorithms, what are the problems, where do the problems come from and how can they be solved.
As was pointed out by Li Xiaosheng, the basics can be better understood by carefully examining the Marching Squares algo first. Actually, the whole answer was layed down by Li Xiaosheng, I've just expanded the explanations a bit.
