You need to be able to recover architectural register state (e.g. if there is a branch misprediction, or an instruction throws an exception). Many implementations store speculative results in a reorder buffer, and write to an architectural register file on commit. The architectural register file always reflects the committed state of the processor. In such an implementation, your renaming table (the "lookup table mapping ISA registers to physical registers") might point to either a ROB entry, if the most recent write to an architectural register was an instruction that is still in flight, or the architectural register file.
Another way to do it is to use a physical register file to store data, as Tuna-Fish explains above. In that case, your front-end table always points into the PRF, but it may be a speculative value (written by an instruction that's still in the ROB and not yet committed) or a committed value. But that doesn't let you recover the architectural state on an exception. So you keep a second table (in the Pentium 4, it's called the RRAT), and whenever an instruction commits, you update that table to indicate that the most recent committed version of a particular architectural register can be found in a particular physical register.
There is at least a third way to do it, which doesn't use an RRAT. MIPS R10k rolls back the ROB to recover the last committed architectural state.
Another way to do it is to use a physical register file to store data, as Tuna-Fish explains above. In that case, your front-end table always points into the PRF, but it may be a speculative value (written by an instruction that's still in the ROB and not yet committed) or a committed value. But that doesn't let you recover the architectural state on an exception. So you keep a second table (in the Pentium 4, it's called the RRAT), and whenever an instruction commits, you update that table to indicate that the most recent committed version of a particular architectural register can be found in a particular physical register.
There is at least a third way to do it, which doesn't use an RRAT. MIPS R10k rolls back the ROB to recover the last committed architectural state.