You can change your name and sex/gender with the Andalusian Health Service (SAS), even if you have not changed them at the Civil Registry. This may be especially useful for people with a non-Spanish nationality who cannot change their name and sex/gender in their country (and also not in Spain with the Trans Law), or for non-binary people or other identities in Spain for which the Trans Law does not apply.
The change of name and sex/gender is regulated by the Procedimiento de la Dirección General de Asistencia Sanitaria y Resultados en Salud del Servido Andaluz de Salud sobre el cambio de nombre de personas transexuales en la Base de Datos de Usuarios de Andalucía (BDU) y emisión de documentos de acreditación (Procedure of the Directorate General of Health Care and Health Outcomes of the Andalusian Health Service on the change of name of transsexual persons in the Andalusian User Database (BDU) and issuance of accreditation documents) from 2016.
This procedure says: "In order to safeguard confidentiality and the right to privacy, these cases will be referred directly to the person responsible for the District, Health Management Area or Hospital Citizen Care Unit, who will provide information, check documentation and provide the corresponding forms, depending on the age of the patient."
This means that an administrative appointment must be made at the Health Center or at the hospital. The procedure document (attached to this text) includes two forms: one for persons over 16 years of age, and one for persons under 16 years of age, who need the consent of their parents or legal representatives.
Non-binary persons
Alana fought four years with the SAS to get their registration with a non-binary sex/gender, in the end successfully.
The forms included in the procedure document do not (yet) allow for a sex/gender to be specified beyond 'male' or 'female'. However, the SAS also allows an 'unspecified' sex/gender, as they have confirmed in a letter to the Andalusian Ombudsman in November 2019, when they also accepted to register Alana with “unspecified sex”. Perhaps it is advisable to take this letter with you when you go to your Health Center, in case they have no idea.
Advice
It is quite possible that your Health Center has no idea that this procedure exists, and that they tell you that it is not possible to change your name and/or sex/gender if it is not on any identification document. You can bring a copy of the procedure with you to explain that nothing is needed - your statement is sufficient.
What happens afterwards?
Your application will go to the Health District, and it may take one or two weeks, maybe a little longer. They will send you a new health card with your chosen name (the sex/gender is not on the health card), and if you have access to Click Salud or have the SAS Salud Responde app, you can see that your name has changed.
The change of sex/gender can only be verified in Click Salud if you have an analytical test, where the sex/gender is included. Or when you have a new blood test or are given a diagnosis sheet.