Does the continuous urethral discharge (madhy) that occurs upon waking from sleep or napping, without a wet dream (ihtilam), necessitate ghusl, especially when one is unable to distinguish between seminal fluid (manyy) and madhy? And would the questioner be sinful if she made an independent judgment (ijtihad) and considered it madhy, or should she perform ghusl to absolve herself of responsibility?

The emission of madhiy (pre-seminal fluid) does not necessitate ghusl (ritual bath); rather, it requires wudu (ablution) and cleansing the parts of the body and clothing it touched. If you are uncertain whether the discharge is mani (seminal fluid) or madhiy, you may choose between them [i.e., apply the rulings for either based on your best judgment]. If you suffer from incontinence of madhiy, you should perform wudu for each prayer after its time has begun. The discharges you see might also be among the normal secretions known as vaginal wetness, which are pure but invalidate wudu.