If you want to focus on the real world, we need to realise that the savings equal investment (S=I) is an accounting identity. That is, it always holds, no matter what.
Importantly, it holds on any time frame, which means that every single transaction will obey it. (You would need to imagine an extremely short-term national accounts that reflects just one transaction.) Therefore, there is no conceivable set of transactions that break the accounting identity, assuming that the accounting is done correctly.
The supply/demand relationships in the real world are trickier. Production of goods could exceed "demand", as the excess ends up in inventories. However, we could pretend that the increase in inventories was planned, and so once again production equals demand. The money supply/demand relationship just tells us that the private sector allocates between "money" and "bonds", but that appears to be another accounting identity. (I am unsure about that statement, and it may take a few steps to get to that point. Since all financial instruments end up being held by some entity, presumably supply and demand have to match.)
In other words, accounting identies always hold, and that is independent of what any particular model says.
If we return the model, "equilibrium" is an assumption/solution technique to find a solution to the systems of equations. That is, within the IS/LM model, the market balance conditions determine the solution. If we change the model, we might get difference conditions to determine the solution.
Since a mathematical model is not the same thing as the real world, we should not say that real world behaviour is directly related to the mathematics. That is, it is unclear if it makes sense to ask whether the real world reaches "equilibrium", without defining what real-world phenemona define "equilibrium." Instead, the test is whether the mathematical model (whose solution is defined by "equilibrium") offers useful insight to real world behaviour.
