Demisexual

Demisexual is a sexual orientation on the asexual spectrum defined as someone who does not experience sexual attraction until they have formed a deep emotional connect with someone.

Demisexual can be a sexual orientation on its own or can combined with other orientations. For example, one could be demisexual and gay, meaning that when one does experience sexual attraction it's only towards people of the same gender.

The romantic counterpart is demisexual is demiromantic.

Etymology
The prefix demi- means half or partial. The prefix is also used in word such as demigod.