What is the Taylor Rule with respect to the equilibrium real interest rate?

The Taylor Rule was developed by Stanford economist John Taylor, first to describe and later to recommend a target nominal rate of interest for the Federal Funds Rate (or for any other target rate chosen by a central bank).

Target Rate = Neutral Rate + 0.5 × (GDPe − GDPt) + 0.5 × (Ie − It)

Where,

Target rate is the short-term interest rate which the central bank should target;

Neutral rate is the short-term interest rate that prevails when the difference between the actual rate of inflation and target rate of inflation and difference between expected GDP growth rate and long-term growth rate in GDP are both zero;

GDPe = expected GDP growth rate;

GDPt = long-term GDP growth rate;

Ie = expected inflation rate; and

It = target inflation rate

Although the equation may seem complicated, it specifies essentially two conditions for changing the target nominal rate of interest (in the U.S., the target Federal Funds Rate):

1) If actual GDP is above "potential" GDP (the level of GDP consistent with full employment), then the Fed should increase the target Federal Funds Rate.

and

2) If actual inflation is above target inflation, then the Fed should increase the target Federal Funds Rate

To your question: the nominal rate of interest is related to the real rate of interest by inflation:

Real Interest Rate = Nominal Interest Rate + Inflation Rate

So, if the Taylor rule suggests that the Fed should increase the Nominal Interest Rate (the Federal Funds Rate), then the short run usage of the Taylor Rule will increase the Real Interest Rate, indirectly. Of course, the Taylor Rule intends to enable the Fed to control inflation, so it would be invoked when inflation is high and hopefully result in lower inflation in the future (which would then bring down the real rate of interest).