The Immigration (European Economic Area) Regulations 2016, which implement EU free movement law in the UK, already recognise same-sex marriages and unmarried partnerships. This will be replicated in the eligibility rules for the EU Exit Settlement Scheme for those EU citizens and their family members arriving before the end of the implementation period. Existing non-EU immigration rules also recognise same-sex marriages and unmarried partnerships.
In terms of the post-Brexit immigration regime, we are considering the options for our future immigration system carefully. We are building a comprehensive picture of the needs and interests of all parts of the UK and intend to develop a system which works in the national interest. In doing so, we will provide for equal treatment of same-sex couples, as we do now and in line with our obligations under the Equalities Act 2010.