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The term monogamish was coined by sex columnist Dan Savage to describe committed relationships in which outside sexual trysts are permitted. While monogamish relationships are not sexually exclusive, they are also not necessarily "open" because the outside relationships tend to be treated as exclusively sexual without further expectations.
Monogamish relationships are not the same as polyamorous relationships or even open relationships. Monogamish relationships can look a lot like monogamous relationships with occasional outside dalliances. The primary partners in a monogamish relationship are generally not looking for other relationships and the outside sexual activity may be seen as an enhancement to the central relationship.