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In almost all cultures and countries on this planet, what you're describing would simply be called bureaucracy and a process that involves a lot of bureaucracy would be referred to as a bureaucratic process. Here's one of the several definitions of this term from Cambridge Dictionary:

complicated rules, processes, and written work that make it hard to get something done

Example sentence (taken from the English Oxford Living Dictionaries):

More than 3,600 staff will be given the chance to influence the way the trust is run by pointing out the unnecessary rules, paperwork and bureaucracy which slow them down.

By the way, the corresponding idiomatic term for bureaucracy would be red tape. And believe it or not, it can be a single word if you properly hyphenate it and use it as an adjective: red-tape procedures. Here's what they say about this expression on Wikipedia:

Red tape is an idiom that refers to excessive regulation or rigid conformity to formal rules that is considered redundant or bureaucratic and hinders or prevents action or decision-making. It is usually applied to governments, corporations, and other large organizations.

labyrinthine

Oxford Living Dictionaries gives the following definition:

1 (of a network) like a labyrinth; irregular and twisting.

‘labyrinthine streets and alleys’

1.1 (of a system) intricate and confusing.

‘labyrinthine plots and counterplots’

‘In the process, he unravelled the labyrinthine means by which a
painting bought by war profiteers and sold to German army looters
found its way into the cultural heart of Britain.’

‘For a show that has the labyrinthine, seemingly nonsensical plots of
a soap opera, that's a real accomplishment.’

‘The country's legendary bureaucracy is as labyrinthine as ever, and
its legal system opaque, with separate laws for foreign and domestic
investors.’

‘The labyrinthine diplomacy and politics of the Italian wars are the
real subject of this painstaking book about what Jem meant to others.’

...

Labyrinthine, through its maze analogy, suggests unnecessary complexity and a process that could be made much more simple.

'Convoluted' might be the word you're looking for. Described by Google as:

(especially of an argument, story, or sentence) extremely complex and difficult to follow.

An example of usage:

"the film is let down by a convoluted plot in which nothing really happens"

Onerous is defined by Oxford Dictionaries as:

(of a task or responsibility) involving a great deal of effort, trouble, or difficulty.
‘he found his duties increasingly onerous’

And by Google as:

(of a task, duty, or responsibility) involving an amount of effort and difficulty that is oppressively burdensome.

Cumbersome would be a simple word to use here. In your context, cumbersome would mean something that is slow or complicated, and therefore inefficient.

Usage example -
'Most of the employees were vexed with their company's cumbersome procedures'

red tape is defined by Merriam-Webster as:

Official routine or procedure marked by excessive complexity which results in delay or inaction.

Merriam-Webster example of red tape in a sentence:

You would not believe the red tape involved in getting the required permits.

Employees complain that this methodology is ??? and therefore it would be tiresome to follow it due to the fact it has many phases.

Rigorous - adjective - from the the Cambridge Business English Dictionary:

• 
  

  detailed and careful:

  
  
  
  
  
  1. I want rigorous financial analysis of the options.
  
  
  2. The selection process is extremely rigorous.
  
  
  
  


• 
  

  strict or severe:

  
  
  
  
  
  1. A rigorous monetary and fiscal policy should encourage efficiency.
  
  
  2. Refrigeration of food, improved hygiene and rigorous standards in the food industry prevent gastro-intestinal infections.
  
  
  
  

Meticulous may be ideal here. It is defined as "taking or showing extreme care about minute details; precise; thorough" (dictionary.com).

Or, perhaps Superfluous would be an appropriate word, meaning "being more than is sufficient or required; excessive" or "unnecessary or needless" (dictionary.com). However, this word would be more effective if describing unnecessary procedures, rather than necessary albeit annoying procedures.

Merriam–Webster gives the second, and more distinctive, meaning of rigmarole as "a complex and ritualistic procedure that is characterized more by form than genuine meaning". It often applies to official procedures that, over a long period of time, have adapted to changing circumstances by accumulation, because no-one has had the authority to adapt them by radical simplification.

One might describe such a procedure as anfractuous:-

characterized by twists and turns; convoluted [Collins English Dictionary via the Free Dictionary]

I believe the American taxation system was once so described by a famous politician as an anfractuousity but sadly can't run down the quote; it must be quite old though as words like that don't fit well in a sound bite. Even the spell-checker here doesn't like it.