I'm afraid it may take a little more investigating. You didn't say whether this happens with the ignition off which should switch the radio off also unless the RAP (retained accessory power) feature is malfunctioning. I believe the radio fuse also powers the antenna signal amplifier located in the headliner above the passenger seat. If so, replacing just the radio probably wont fix it.
Not sure what to tell you to try. That radio fuse is 20A if I recall. I might try putting a much smaller fuse (like a 2A) in place of the radio 20A and see if it blows when the radio is off. The signal amplifier likely does not use much power so it would need to be a very small fuse.
You didn't say what the age/condition of your battery is. The RL (and most Honda's) is very finicky about quality battery power. if you are having problems and the battery is over 2 years old, I would consider replacing it