Imagine that Général Bonaparte does something smart: noticing that there is no Suez canal yet, he does not waste an army in Egypt. Instead, according to a secret clause in one of the Franco-Spanish treaties, the French expeditionary corps helps the Spanish take Gibraltar. For example, the corps embarks at Toulon, hugs the coast and disembarks at Malaga. The French army is good at manoeuvre and artillery and manages to catch the British mostly off-guard and take the fortress. (In effect, the POD has saved the Armée d'Orient, the squadron, and Kléber, and has made them do something useful of themselves.)
Now the Rock itself is quite safe for the time being: it is probably out of direct reach from the British Isles, and even if the British manage to bully Portugal and use Lisbon as a stepping-stone, the fortress is likely to be extremely well defended by the Franco-Spanish. As the French come as genuine Spanish allies, they will not experience the guerilla problem. Spain can also probably be the bigger of the two bullies as far as Portugal is concerned...
So the British are cut off from the Mediterranean for some time. The Minorca garrison, if already present, is isolated; let's say it manages an heroic escape, by night, and evades to Lisbon and later Britain (like OTL Denis Decrès escaped from Malta, except the British have better ships and captains).
The other victim will probably be Austria: while the sea lanes between Britain and Austria are cut, the Armée d'Italie will enjoy friendly seas at its back; in particular, Masséna will welcome the relief in besieged Genoa. Moreover, Corfu is in French hands at the time, and probably quite safe without British support, as long as the French don't aggravate the Turkish too much (OTL the French garrison was expelled by a joint Russian-Turkish attack, which is a diplomatic almost-ASB; if the French don't do something stupid such as attacking Egypt, there is no way the Turkish let a Russian fleet through the Bosphorus); the French forces there are quite minor, but could be a minor nuisance in the Adriatic (again, to the great pleasure of the Grand Turk).
Assuming the French can keep the Turks content, the main nuisance will be the Barbary pirates; instead of an Egyptian Expedition, we could see, a few years later, an Algerian one, as both the French and Spanish have interests here. Probably not outright conquest (too soon, and manpower is needed in other places) but maybe the capture of a few key ports/fortress, if only to deny their use to the British. As this almost coincides with the Barbary War, the Franco-Spanish will find themselves almost co-belligerent with the United States...
Or maybe Spain instead tries to grab the old Aragonese possessions of Naples, Sicily or Sardinia. Assume Sardinia as it is probably the easier target, being the less populated, and continental Savoy being under French occupation. As Sardinia is under raiding from the Barbary pirates at the very same time, this could actually be the spark that decides for the Barbary expedition of the above paragraph...
The Treaty of Lunéville will probably happen, and may be slightly more pro-French in Italy. Maybe Austria does not get Venice: the Republic is instead turned into an Austrian buffer state. So North Italy is split in two buffer zones, without either direct French or Austrian control; it is slightly less of a powder keg than OTL. Southern Italy is probably at least under Spanish influence. Austria is compensated in Illyria (and Corfu if the French are reasonable). North of the Alps, the Rhine is the border as OTL.
The treaty of Amiens is slightly more complicated, but a basic “English out of Gibraltar and (random Spanish colony)” vs. “Franco-Spanish out of Portugal (and maybe Netherlands)” deal is possible.
On a longer term, the French navy is able to use the Mediterranean as a training ground and partially solve its crew problem. As in OTL, the Marine Nationale has a few correct ships and shipyards and some competent admirals, not enough to be fully on par with the Royal Navy, but probably enough to be, locally, a serious opponent. So the situation seems stable, with the British controlling the Atlantic, and the French and Spanish controlling the Mediterranean.
Any long-term French ambitions will probably be set on (northern) Italy. The French can implement a lightweight Continental system; if we want to remain in the theme, we can call it the Mare Nostrum system. (btw, this sounds like a cool TL name. Dibs on it.) The British Isles themselves are not threatened. If the British are angry at French meddling in Italy and declare war, it would still take an extraordinarily stupid French ruler to try and invade Britain; note that this does not mean that they will not try...