How to check Einstein-like equations on their correspondence with the real world?

Considering all the possible low-order corrections to Newtonian motion that there might be to a particle moving in a gravitation field of any kind, or rather, in comparing relative motions of neighboring particles, there are defined some paramaters according to the Parameterized Post Newtonian (PPN) formalism. This is described nicely in the famous book by Misner Thorne & Wheeler. (My other three books on GTR don't mention PPN.)

GTR can be analyzed in terms of PPN parameters - beta=1, eta=0, etc. Alternative theories work out to having different PPN values. Any experimental observations, after being calibrated and curves fitted to and so on, can be judged in terms of PPN parameters, at least some of them, or combinations of some of them. So it's a matter of seeing what fits within error bars.

http://en.wikipedia.org/wiki/Parameterized_post-Newtonian_formalism tells more detail.

Physical theories are rarely constructed as a purely theoretical exercise - they usually are based on empirical data which is fitted a mathematical solution. The physical theory then tries to explain the data and corresponding math.

The basis of scientific theory is that it must be verified by repeatable tests and data. Sometimes theories that explain some data well are disproven when applied to other situations - and sometimes amendments are made in order to account for the difference. That is the nature of every scientific theory - it is accepted until real world data says it is wrong. There is no way to know whether any theory truly describes our world or not - it holds until it no longer matches observations.