Basic Trope: Two government organizations (often branches of the military) have a rivalry that may or may not affect their work.
Straight: The Army and Navy of Troperia have a rivalry.
The Army and Navy of Troperia regularly exchange fire.
The web of rivalries is very confusing: the Army paratroopers have a good relationship with the Marines, but they both hate the regular Army infantry. Meanwhile, the Marines get along well with Army aviation out of mutual dislike for the Navy's carrier-based fighter wings, who for some odd reason have a grudge against Air Force intelligence (but surprisingly, not the fighter pilots...)
Downplayed: The Army and Navy of Troperia have a friendly rivalry, but have no ill will towards each other.
Justified: The Army and Navy have different cultures and missions, so naturally there's posturing going on about which service is "better". Not to mention that they compete for government funding once budget time rolls around.
Inverted: The Army and Navy hold special ceremonies in honor of each other and have perfect battlefield coordination.
Subverted: The rivalry turns out to be just a planned joint-act by the friendly Army and Navy leaders to draw the attention of the Troperian president.
The concerned Troperian president raises the funds of the Army and Navy to mitigate the alleged problems... but now the Air Force is madly jealous and annuls any further collaboration with the the other branches.
The Army and The Navy use their new funds to create amphibian divisions that collide so much with each other that the two branches become enemies.
Parodied: The heads of the Army and Navy were schoolmates who hated each other.
Zig Zagged: The Army and Navy have a rivalry, but they both get along well with the Air Force. All three of them absolutely hate the Strategic Missile Forces... which takes a neutral stance towards to all of them.
Averted: The Army-Navy rivalry is never brought up, or doesn't exist.
Enforced: The producers of Troperia Land felt that the show gives an idealized image of the military so the writer was asked to spice it up adding more conflict.
Lampshaded: "What are you doing? We're on the same side here!" "No we're not, you're in the Army!" "Navy pukes!"
In order to gain political influence, the Air Marshall spreads malicious rumours hoping to antagonize the Army against the Navy. It works and he becomes the new unified commander when the other two leaders are sacked.
The paranoid President of Troperia encourages all the powerful factions within the country — The armed forces, State Sec, police, various intelligence agencies — to be distrustful of each other to prevent anyone rising up against him. Meanwhile, he increases the power of his Praetorian Guard until they're effectively a private army.
Exploited: The top military staff of Ruritania exploit the Interservice Rivalry of the Troperian military to Divide and Conquer. With the various arms of the Troperian military at odds with one-another and thus unable to effectively counter the devastating combined assault, Ruritania easily marches in and takes control.
Defied: The Field Marshal and the Marshal of the Air Force have always been bitter rivals, but their immediate subordinates get along with each other much better. The subordinates frequently go against the orders of their superiors and focus on winning engagements rather than taking risks over a petty rivalry.
Discussed: "So, have you heard the Field Marshal and Marshal of the Air Force have been arguing again?" "Yeah. If they keep this up, the Army is gonna get hammered by Ruritanian close air support. We have to try and get them to agree to something!"
Conversed: Alice and Bob are talking about their married landlords and they point out they are like those military honchos in the movies dressed in distinct colours, always bickering and uncooperative.
The mutual distrust between the services leads to the loss of a crucial battle because the two services were following different plans and refusing to support each other. The war, which might have been ended in weeks, drags on for years.
The multiple power blocs of Troperia devolve into Chronic Backstabbing Disorder, making it easier for their enemies to defeat them one battle after another.
Reconstructed: The rivalry between the services leads each branch to come up with diverse and tactics and equipment meant to one-up the other force. When war comes, this gives Troperia an edge over Ruritania, whose monolithic military organization controls all three branches and stifles creativity and initiative.
Played For Drama: The Army-Navy rivalry hampers cooperation, and a lot of people who shouldn't have died do because Army and Navy weren't cooperating.
Played For Laughs: The Army and the Navy are literally tripping over each other to attack an enemy position to the point that they fail do do any real damage whatsoever. The enemy stops fighting them to watch and popcorn is distributed.
Back to Interservice Rivalry, but watch out for those Navy creeps. They're as sneaky as they are wimpy.