One of the most fascinating and least widely appreciated facts about experimentally verified fundamental reality is that the vacuum spacetime that we inhabit is densely filled with topologically nontrivial principal SU(n)-bundles. This is a fact confirmed by matching experimental results against lattice QCD computations. For a detailed review see section 5 and the illustrations in section 7 of
"Topology in dynamical Lattice QCD simulations",
Also see the lattice references listed in the introduction of
"QCD and the eta prime Mass: Instantons or Confinement?",
Phys.Rev. D67 (2003) 074502 http://arxiv.org/abs/hep-lat/0211035
and for more references see athttp://ncatlab.org/nlab/show/instanton+in+QCD
Namely the fields of the weak and strong nuclear force are mathematically modeled by connections on SU(n)-principal bundles for n=2 and n=3, and the moduli space of such connections globally shows "topological twists" measured by the second Chern-class of these bundles.
(See "Higher field bundles for gauge fields" for more http://ncatlab.org/schreiber/show/Higher+field+bundles+for+gauge+fields
The nature of these topological twists is of a similar nature as the knottedned of knots. Where a nontrivial knot is a map from a string into a 3-sphere that may not be isotypically shrunk to a point, so a nontrivial gauge field connection is a smooth map from spacetime into a certain moduli stack, and it is called "topologically twisted" if it may not smoothly be deformed to a constant map.
In physics these topological non-trivial gauge field configurations are mostly kown as "instantons", and hence the second Chern class of an SU(n)-principal bundle is in physics also known as the "instanton number". Notice however that this terminology is overloaded: strictly speaking an instanton configuration is a topological nontrivial field configuraton plus a certain condition on minimality of the Euclideanized energy of the configuration. But the second Chern class is insensitive to this additional constraint on the energy, and for the purpose of lattice QCD, all that one cares about regarding instantons is this topological effect. So in particular, despite the origin of the term, this is unrelated to and insensitive to any "Wick-rotating" the spacetime signature from Lorentzian to Euclidean signature.
Nontrivial topological charges of nuclear gauge fields in the vacuum are real.