Well, it’s hard to say.  I’m assuming it is dripping right out of the carburetor throat when tilted, but have you confirmed that?  Does it start to drip as soon as the engine is tilted upward?  I suppose there is the remote possibility that the bowl gasket/surfaces are messed up causing the leak, but not very likely.  Perhaps the float is set just a bit too high.  Perhaps something is pressurizing the fuel tank overcoming the inlet needle/seat.

In any event, sounds like you will be back inside this engine soon for the powerhead swap.  So, I would recommend pulling the fuel line and letting most of the fuel run out of the carb while running, then shutting it down.  Yes, I have heard the argument saying that running the engine “dry” will result in rusted internals.  In theory this is true, but having the engine sit for a few days like this isn’t going to hurt anything.  Running the engine out of fuel this way doesn’t even come close to drying out the crankcase, there is still plenty of oil on the internals.

Food for thought:  How could the internals possibly run dry of oil when running out of fuel?  It just can’t happen.  When it runs out of fuel (and oil) it quits running.  And what happens if you run the tank out of gas at 4500 RPM?

Having said that, OK, OK, speaking of a single carb motor here.  I suppose there is some argument that on a multi-carb motor, one carb could run dry before the other.  That probably is how the story got started.

The float and float needle are designed to work with the motor in approximately a vertical position. When the motor is tilted upwards, the float and needle are in a less efficient position. When the fuel line is connected, and there is pressure in the line, from agitation, sun, fuel pump, or pressure tank, maybe fuel is slipping past the float needle.

As others have stated, maybe running the carburetor out of fuel, but more importantly, disconnecting the fuel line is the key.