I can only thing of 2: F-15E+, and F-16 Block 30+ can take either F100-220/220E/229 or F110-100/129. There have been others where the production engine changed: F-86 got an Orenda engine for later Canadian (maybe Australian also), F-4K got RR Spey. There may be others, but very few have had interchangeable alternative engines.
A few thoughts - as distinct from licensed production and additional engineering (Rolls & Pratt with the J48; Armstrong Siddeley and Wright with the J65), second source manufacturing for jet engines was relatively common during the 1950s - some examples in the US being:
GE J35 – Allison and Chevrolet
GE J47 - Packard and Studebaker
P&W J57 – Ford
Wright J65 - Buick
(which makes sense with the B-47 needing six engines and the B-52 needing eight.)
I believe that parts were interchangeable between manufacturers for the particular engine variant, but would have to dive into the specific IPB catalogues to verify - I'll leave that as an exercise for the reader.
Aircraft production numbers declined in the sixties and seventies which allowed OEM production only. The last attempt to second-source an engine was when the Navy split the buy of F404s between GE and Pratt with the F-18 being the intended application (this was in addition to the anticipated production of the A-6F with dry F404s). A production contract for 102 engines worth $174m was awarded to Pratt in FY88. This was another watermark of the Lehman era as SecNav.
I think that the intent was that the Pratt engines be Form/Fit/Function drop-in replacements for the GE-built engines. In the event, this was not to be; for whatever reason (and there was a lot of blamestorming) Pratt parts were not interchangeable with GE parts, nor did the engines perform exactly the same (differing stall margins, etc). In addition, costs were higher and the Pratt program ran late as they struggled to replicate GE processes based on the Navy reprocurement data packages. Eventually, the A-6F was cancelled which relieved the production pressure on GE; the lowered production rates then rendered the second source completely uneconomical (along with Pratt's production difficulties).
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I should note that I am using the term "second source" to mean an engine that is license built by another manufacturer than the OEM and is interchangeable in the target aircraft. The use of a different engine for the same application, I would call an "alternate engine" such as the F135/F136. This is quite common in the airliner field, particularly with Boeing in the 70s and 80s. IIRC, the 747-400 and the 767 were certificated with the choice of three different manufacturers engines.