In vacuum and with only the particles we know about the answer is no. Let's look at the symmetries we know exist in nature:
- $SU(3)$ colour: confined, only colourless states exist below the QCD phase transition
- $SU(2)\times U(1)_Y$ electroweak: Higgsed to $U(1)_{EM}$ electromagnetism
- $U(1)_{EM}$: Here we have opportunity. See below...
- $U(1)_{B-L}$: Global symmetry in SM, possible gauged symmetry of GUT or not a true symmetry at all. If gauged it's broken at a high scale. Note that we don't see any massless Goldstone bosons for this symmetry, so it can't be a spontaneously broken global symmetry. It must be Higgsed or not a symmetry at all. In either case breaking has already happened or never will. Breaking of this is relevant to baryogenesis but not to much else.
- QCD chiral flavour symmetry: only an approximate symmetry. Broken by chiral condensates, the pseudo-Nambu-Goldstones are the mesons. (I'll stop listing approximate symmetries otherwise this list will get very long, but this one is rather important.)
- Local Poincare invariance: Exact and "gauged" in general relativity. (Note: there is an ongoing debate about the semantics of whether gravity is a gauge theory. There are important similarities and differences between gravity and the standard Yang-Mills gauge theories. Hence the scare quotes on "gauged.")
- Global Poincare invariance ($SO(1,3)\ltimes \mathbb{R}_4$):
Spontaneously broken by the fact that the universe is expanding and there is stuff in it. This is a symmetry of Minkowski spacetime, so it is often used in particle physics, but it is not a symmetry of our actual universe because it is expanding. In general there are no global symmetries or conservation laws in GR, but the usual spacetime symmetries hold to a very good approximation on galactic cluster and smaller scales. My previous language meant to convey this but was sloppy and inaccurate.
As far as I can see there are two options for spontaneous symmetry breaking in the current universe: either attack #3 or #6. What do you need to have to break either of these groups? You need an order parameter that transforms nontrivially under the symmetries to take a nonzero expecation value.
For electromagnetism that means you need a charged condensate, but we don't know of any charged scalars and the chiral condensate is necessarily neutral (why is that? good question ;)). In principle one of the $W^\pm$ could serve (spontaneously breaking Lorentz invariance as well), but this can't happen because they have large positive mass squared through the Higgs mechanism. You would have to generate a negative effective mass squared using some fancy new mechanism that definitely doesn't exist at the low energies we can see. So you can't break EM in vacuum, but you can in a medium where collective motions of many particles serve as the condensate. These exotic materials are called superconductors, and a few people think they are mildly interesting. ;)
That leaves local Poincare invariance. This can be broken by a vector or tensor field developing a condensate. People have looked at these sorts of models, but needless to say there is nothing like this in known physics. Experiments have demonstrated Poincare invariance to an incredible accuracy. Given the accuracy of the experiments and the cosmological scale of the transition temperature we're talking about you would need a vector or tensor with a cosmologically small negative mass squared. Needless to say this is problematic, especially if you want to identify these with the known gauge bosons or graviton.
