What is the ruling regarding someone who experiences a wet dream but does not perform ghusl out of embarrassment towards his colleagues, and how can he perform ghusl in the wilderness without feeling embarrassed by people?

Firstly: Whoever is in a state of ritual impurity (janabah) and finds water, must perform ghusl (ritual bath). If he does not find water, or fears harm from using it due to extreme cold, he should perform tayammum (dry ablution) and pray. It is not permissible to abandon ghusl and prayer out of shyness or embarrassment.

Secondly: It is not necessary to undress completely during ghusl; rather, it is recommended to cover the 'awrah (private parts) with an izar (loincloth) or similar garment, even when alone. Covering is better than uncovering. So, whoever is in the wilderness and wishes to perform ghusl, should wear an izar or cover himself with a garment and perform ghusl away from people's sight, or while sitting such that his 'awrah (the area between the navel and the knee) is not visible.

Thirdly: Whoever neglects ghusl in this situation and performs tayammum while water is available or he is able to perform ghusl without harm, his tayammum is invalid, and he must repeat the prayers he performed before performing ghusl.