I'm not sure that 'authentic' means that people respect each others' beliefs.
To me it means that they are open with their own beliefs, and stay true to their own beliefs, but doesn't necessarily imply that they respect the beliefs of others.
I'm open to correction on that though.
The expression that immediately springs to mind is open-minded .
An open minded person is an unprejudiced person who is receptive to new and different ideas or the opinions of others, and is always willing to listen to and hear an opposing or contradictory view.
ell.stackexchange.com/users/44922/mike
When applied to a person (or someone's personality) authentic means that they are "true to themselves" -- that their words and deeds accurately reflect the real person.
For example, Bernie Sanders (US Senator and presidential candidate) is often considered to be authentic because his speeches and campaign promises are a reflection of the things he has supported and promoted his entire political career.
However, authentic does not in any way imply tolerance or open-mindedness.
You could very easily call someone "an authentic asshole" if you thought it appropriate, meaning that they are honestly as much a jerk on the outside as they are on the inside.
You may be looking for sincere , or one of its many synonyms .
ell.stackexchange.com/users/42184/Andrew
Someone who has integrity has 1. firm adherence to a code of especially moral or artistic values 3.
the quality or state of being complete or undivided They have beliefs they adhere to.
They are "undivided": their actions are not opposed to their beliefs.
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It sounds like you might be looking for honorable or one of its synonyms, such as upstanding , ethical , or conscientious .
Given the emphasis on having and upholding a personal set of beliefs, I think principled might work well for you.
From Cambridge Dictionaries : adjective US ​ /ˈprɪn·sə·pəld/ based on principles, or (of a person) having good personal standards of behavior: She was known among her colleagues as a principled professional.
So People [who respect each others' beliefs as well as uphold their own beliefs and systems] are tolerant and principled .
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There are various descriptors for people who strongly believe in whatever "their belief system" is true believer fundamentalist - maintains core beliefs advocate - promotes a belief gate
keeper - maintains the standards of a belief dyed-in-the-wool - can not change their belief some of these may also have religious overtones, and have no explicit meaning in terms of "open mindedness", though usually if one has a very strong belief in something it may limit inclusion of other belief systems.
In your examples a traditionalist is someone who believes in the "tradition" of "how things were or have been".
Where as authentic means "genuine" without regard to the strength of belief.
Usually, additional context is necessary to bring together the disparate sides of "strong belief" and "inclusion"
inclusive open mindedness sympathetic
He is a true believer in the free market system but still sympathetic to the social safety
net
They advocate democracy while still respecting the monarch.
She is a dyed-in-the-wool Democrat while still wanting tax breaks.
ell.stackexchange.com/users/26439/Peter
I want to express that people respect each others' beliefs as well as uphold their own beliefs and systems. Tolerant means respecting others' beliefs, but what word stands for "uphold their own beliefs"?  Does  "authentic"  convey this meaning clearly in the following sentence? What are the other alternatives? Authentic person generally means one who is not fake.   People are tolerant and authentic.   Does  "traditional"  work here? Does it have any negative connotations?
Correct sense of the word "authentic"
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<word-usage>
ell.stackexchange.com/questions/111181
ell.stackexchange.com/users/38086/emmy
Many terms may be used to convey your intended meaning, including genuine, true to themselves, undivided, sincere, principled, open-minded, and  sympathetic. However, this depends on the context. "Authentic" means beaing true to oneself, genuine, but does not reflect the concept of open-mindedness. Some words that could work to express this concept are: "sincere", "undivided", "honorable", "upstanding", "ethical", "conscientious", principled", "inclusive", "open-minded", "sympathetic". However, additional context is needed for a term that includes all aspects of these concepts. As far as "traditionalist" is concerned, it means someone that believes in "the way things once were".
"Authentic" means beaing true to oneself, genuine, but does not reflect the concept of open-mindedness. Some words that could work to express this concept are: "sincere", "undivided", "honorable", "upstanding", "ethical", "conscientious", principled", "inclusive", "open-minded", "sympathetic". Additional context is needed for a term that includes all aspects of these concepts. "Traditionalist" means someone that believes in "the way things once were".