The hypothetical particles you refer to are known as tachyons, you can look them up for more information.
My personal interpretation of the equations is that these solutions are meaningless. After all, say a cubic equation comes up in some physically relevant problem and there is only one real root. Do we say there exists another universe, separate but disconnected, in which the two complex roots are manifest in some physical form? No, almost anyone would say that's nonsense. My opinion of the equations of relativity is analogous.
But just to give you a bit of information about those who do take the possibility (somewhat) seriously, your description is correct. However, these objects cannot interact with our physical reality. This includes being able to detect them in ANY way (of course, this necessarily must involve interaction!). If they could, causality would be violated and I do not think the physics community at large will ever be willing to throw that out the window unless there is overwhelming evidence to support such a conclusion.