Word for Secretive Marriage that is not Elope

Solution 1:

You might call it a clandestine marriage

adjective
kept secret or done secretively, especially because illicit.
"she deserved better than these clandestine meetings"

(oxforddictionaries.com)

The wikipedia article on a Fleet Marriage, for example, refers to clandestine marriages as a type of fleet marriage:

..."Clandestine" marriages were those that had an element of secrecy to them: perhaps they took place away from a home parish, and without either banns or marriage licence

For another example (that this is not just [synonym for secret] stuck in front of 'marriage'), the subject and title of this book:

(Google Books)

Solution 2:

These two people have a private marriage.

1.1 (Of a conversation, activity, or gathering) involving only a particular person or group, and often dealing with matters that are not to be disclosed to others:
this is a private conversation a small
private service in the chapel
Oxford Dictionaries Online

Solution 3:

Such a marriage could be considered, in modern parlance, to be on the “down-low.”

down-low: (also "on the down low" or "on the DL") may refer to any activity or relationship kept discreet. Specifically, it may refer to:

• Keeping an act, action or some other piece of information a secret.

• Down-low (sexual slang): Men who identify as heterosexual, but have sex with men secretly.

(Wikipedia)

down-low noun (uncountable):

1. secrecy I'll tell you, but keep it on the down low.

2. (sexuality) the state of being a man who secretly sleeps with people other than his partner

3. (sexuality) the state of being a man who secretly sleeps with other me.

(wiktionary)

Solution 4:

Since most members here are middle-aged, I presume, I suggest an old slang term:"on the QT"

• The slang term 'qt' is a shortened form of 'quiet'. There's no definitive source for the phrase 'on the q.t.', although it appears to be of 19th century British origin - not, as is often supposed, American. The longer phrase 'on the quiet' is also not especially old, but is first recorded somewhat before 'on the qt', in Otago: Goldfields & Resources, 1862

• "on the Q.T. (alternative forms: on the qt, on the QT, on the q.t.) (idiomatic) Quietly; in a secretive manner; clandestinely. Wiktionary

e.g.

• They told her on the Q.T. that she was being promoted.
• They got married on the Q.T.