Nach unserem Streit gestern haben wir heute hart gearbeitet und trotzdem zusammen gelacht.

Because nach is a preposition that always takes the dative case.

• Streit is masculine:
  
  • Nominative: der Streit
  • Dative: dem Streit
• With the possessive unser-, the dative masculine ending is -em, so:
  
  • unser + dem Streit → unserem Streit

So nach unserem Streit literally means after our argument with our argument in the dative case.
unseren Streit would be accusative (e.g. Wir erinnern unseren Streit – We remember our argument), which is not correct after nach.

In the sentence:

Nach unserem Streit gestern haben wir heute hart gearbeitet und trotzdem zusammen gelacht.

• gestern refers specifically to Streit → the argument that happened yesterday
• heute refers to the actions haben … gearbeitet and (haben) … gelacht → today we worked / laughed

You can move them, but the meaning and focus shift slightly:

1. Gestern nach unserem Streit haben wir heute hart gearbeitet …
   This is unusual, because gestern and heute clash at the start; you’re saying Yesterday after our argument we have today worked… – not idiomatic.

2. Nach unserem Streit gestern haben wir hart gearbeitet und trotzdem zusammen gelacht.
   Now heute is gone; it sounds like After our argument yesterday, we worked hard and still laughed together – all on the same day as the argument.

3. Nach unserem Streit haben wir heute hart gearbeitet und trotzdem zusammen gelacht.
   Now gestern is gone; we don’t know when the argument happened, only that today we worked and laughed.

So the original version is a neat way to say:

• the argument was yesterday
• the working and laughing are today

German main clauses follow the verb-second rule (V2):

• The conjugated verb must be the second element in the sentence.
• The first element can be the subject, a time expression, a prepositional phrase, etc.

In your sentence:

1. Nach unserem Streit gestern = first element (a prepositional phrase of time)
2. haben = second element (the conjugated verb)
3. wir heute hart gearbeitet und trotzdem zusammen gelacht = the rest of the clause

This is why you don’t say:

• ✗ Nach unserem Streit gestern wir haben heute …

That would break the V2 rule.
Compare with a subject-first version:

• Wir haben nach unserem Streit gestern heute hart gearbeitet und trotzdem zusammen gelacht.

Now wir is first, haben is second – still V2.

trotzdem and obwohl both express contrast, but they work differently in the sentence:

• trotzdem = nevertheless / despite that → an adverb in a main clause
• obwohl = although → a subordinating conjunction introducing a subclause

Your sentence uses trotzdem:

Nach unserem Streit gestern haben wir heute hart gearbeitet und trotzdem zusammen gelacht.
After our argument yesterday, we worked hard today and still / nevertheless laughed together.

To use obwohl, you change the structure:

Obwohl wir gestern Streit hatten, haben wir heute hart gearbeitet und zusammen gelacht.
Although we had an argument yesterday, we worked hard today and laughed together.

Key points:

• trotzdem does not start a subordinate clause; it stays in a main clause with normal V2 word order:
  • Wir hatten gestern Streit, trotzdem haben wir heute zusammen gelacht.
• obwohl starts a subordinate clause, and the verb in that clause goes to the end:
  • Obwohl wir gestern Streit hatten, …

So yes, you can express a similar idea with obwohl, but you have to restructure the sentence.

trotzdem is a sentence adverb; it belongs in the Mittelfeld (the middle field of the sentence) and is somewhat flexible, but not everywhere is equally natural.

Natural options:

1. Nach unserem Streit gestern haben wir heute trotzdem hart gearbeitet und zusammen gelacht.
   → trotzdem modifies both actions: worked hard and laughed.

2. Nach unserem Streit gestern haben wir heute hart gearbeitet und trotzdem zusammen gelacht.
   → The contrast is felt especially before laughing together.

3. Nach unserem Streit gestern haben wir heute trotzdem zusammen hart gearbeitet und gelacht.
   → Emphasises “even so, we worked together hard and laughed”.

Also possible:

4. Trotzdem haben wir heute nach unserem Streit gestern hart gearbeitet und zusammen gelacht.
   → Starting with trotzdem is also fine; verb (haben) still stays in second position.

Unnatural/wrong:

• ✗ … haben wir heute hart gearbeitet und zusammen gelacht trotzdem.
  Putting trotzdem at the very end sounds wrong in standard German.
• ✗ … haben wir heute hart gearbeitet und gelacht trotzdem zusammen.
  Also very odd.

So the sentence’s original position is natural and common, but you do have some flexibility as long as verb-second and normal word order are respected.

Two points: grammar and word choice.

1. Grammar (no ending on hart)
   In hart gearbeitet, hart functions as an adverb, not as an attributive adjective before a noun.
   Adverbs in German do not take endings, so:

• Er arbeitet hart. – He works hard.
• Sie hat hart gearbeitet. – She worked hard.

You’d only add an ending before a noun, e.g.:

• die harte Arbeit – the hard work

2. Word choice (hart vs stark)
   The idiomatic collocation is hart arbeiten = to work hard.
   Using stark arbeiten is either wrong or would sound very unusual. stark is used more for strength or intensity of something else:

• starker Kaffee – strong coffee
• starker Wind – strong wind
• stark verbessert – greatly improved

So hart arbeiten is the natural, standard phrase for to work hard.

In the Perfekt tense, most verbs take haben as the auxiliary. sein is used mainly for:

• verbs of movement / change of place (gehen, kommen, fahren, laufen, etc.)
• verbs indicating a change of state (einschlafen, sterben, wachsen, etc.)
• a few special verbs (bleiben, sein, passieren)

arbeiten and lachen:

• do not express a change of location
• do not express a change of state
• are just regular activities

So they take haben:

• Ich habe gearbeitet. – I have worked.
• Wir haben gelacht. – We (have) laughed.

✗ Wir sind gearbeitet and ✗ wir sind gelacht are ungrammatical.

Yes, you could say:

Nach unserem Streit gestern arbeiteten wir heute hart und lachten trotzdem zusammen.

Grammatically, that’s fine. But in modern spoken German, people usually prefer the Perfekt (haben gearbeitet, haben gelacht) for completed past actions, especially in conversation.

Rough rule of thumb:

• Spoken German: mostly Perfekt
  • Wir haben heute hart gearbeitet und zusammen gelacht.
• Written / formal narrative (stories, reports, novels): more Präteritum (simple past)
  • Wir arbeiteten heute hart und lachten zusammen.

So your original sentence using haben gearbeitet and (haben) … gelacht sounds natural and conversational.

German allows one auxiliary to serve for several past participles in a coordinated structure.

Your sentence:

… haben wir heute hart gearbeitet und trotzdem zusammen gelacht.

Here:

• haben is the single auxiliary for both gearbeitet and gelacht
• Structure: haben
  • [subject + adverbs] + gearbeitet und … gelacht

If you want, you can explicitly repeat haben, but it sounds heavier:

• … haben wir heute hart gearbeitet und haben trotzdem zusammen gelacht.

That’s grammatically correct, but the original version (only one haben) is more natural and fluent.

Word order in the Mittelfeld (middle of the sentence) has some flexibility, but certain combinations are more natural because they form a strong collocation.

• zusammen lachen (to laugh together) is a very common phrase.

• In the Perfekt, the past participle goes to the end, but zusammen tends to stay close to the verb it belongs with, before the participle:

  • Wir haben zusammen gelacht. – Natural
  • Wir haben alle zusammen gelacht. – Also natural

You can say:

• Wir haben gelacht, zusammen.

But that sounds like an added afterthought: We laughed – together, with special emphasis on the together part. It’s possible, but not the neutral, default order.

So zusammen gelacht is the standard, neutral word order.

Streit is a noun from the verb streiten (to argue, quarrel). Its meaning sits between argument, quarrel, and fight, usually in a verbal/emotional sense.

• It often implies raised voices, disagreement, tension:
  • Wir hatten einen Streit. – We had an argument / a fight.
• It does not automatically imply physical violence. For a physical fight you’d more often say:
  • Kampf, Schlägerei, or use verbs like sich prügeln.

So in Nach unserem Streit gestern …, a very natural translation is:

• After our argument yesterday …
  or
• After our fight yesterday …

with the default assumption that this is a verbal conflict unless context suggests otherwise.