Green’s Dictionary of Slang

1. (US Und.) connection, association.

1929		C. Coe Hooch! 120: What’s the tie-up between Dopey Hillar an’ Swinnerton?
1937		W.M. Raine Cool Customer 203: What about the tie-up between West and Garside?
1942		C. Sandburg letter 12 Mar. in Mitgang (1968) 410: I would probably have told him that your own tie-up with him could very well be like my own.
1951		P. Whelton Angels are Painted Fair 222: What bothers me now is the tie-up Mrs. Reutler had with that crew.
1970		‘Red’ Rudensky Gonif 96: Who they were or what their tie-up was didn’t matter.
1986		D. Barker Life in Jazz 52: It is a very interesting and very clever tie-up. The doctor, undertaker, commissary of the society [...] and the brass band leader were all in cahoots.
1992		A. Duff One Night Out Stealing 30: So there’s no connection, no tie-up with Wellington from their end.

2. (US) an emotional/sexual relationship.

1953

F. Brown Madball (2019) 124: Not that he wanted a permanent tie-up with any of these cheap carney broads.