My understanding of real physical theory of electromagnetism goes like this:
The Maxwell equations can be used to derive the speed of light; $$\nabla\cdot\textbf{E}=0$$ $$\nabla\cdot\textbf{B}=0$$ $$\nabla\times\textbf{E}=-\frac{\partial\textbf{B}}{\partial t}$$ $$\nabla\times\textbf{B}=\mu_0\epsilon_0\frac{\partial\textbf{E}}{\partial t}$$ gives $$c=\frac{1}{\sqrt{\mu_0\epsilon_0}},$$ and Special Relativity derives from the fact that this $c$ is constant in all reference frames.
Suppose there was another light-like field $$\nabla\cdot\hat{\textbf{E}}=0$$ $$\nabla\cdot\hat{\textbf{B}}=0$$ $$\nabla\times\hat{\textbf{E}}=-\frac{\partial\hat{\textbf{B}}}{\partial t}$$ $$\nabla\times\hat{\textbf{B}}=\mu_0\epsilon_0\frac{\partial\hat{\textbf{E}}}{\partial t}$$ $$\hat{c}=\frac{1}{\sqrt{\hat{\mu_0}\hat{\epsilon_0}}},$$ with $\hat{c}\neq c$
Suppose we require $\hat{c}$ to be constant in all reference frames. Is it still possible to construct Special Relativity consistently with these two different speeds of light-like fields?
EDIT -- to clarify, based on the discussion under Eric Smith's answer: the entire mathematical structure of a hypothetical universe is up for grabs here. Make no assumptions about reality beyond what is necessary to support the premise. (For example, if this yields a universe in which the definition of simultaneity depends not only on your reference frame but on which type of particle you are, that doesn't have to "make sense", the only issue is whether the mathematics can be consistent in any universe.)