I'm the founder of my current startup. The new startup is an opportunity to leverage my industry expertise twice. I would get shares in the new startup, but not have control. Obviously the idea would be to create a close partnership between both startups (they are quite complementary). I'm afraid of two things: 1/ This new startup may become a direct competitor one day. The industry is very big, with the largest player having only about 5% of the market. Still, our two startups are two of the very few players with an innovative concept in the industry. 2/ The insights that I transfer to the new startup, may spill over to the rest of the industry more easily (as I have little control), reducing our competitive advantage. How could I craft this partnership so that my two fears can never materialize?
Once you have your strategy figured out, make sure you speak with a lawyer to protect you. I am not a lawyer but will give you some advice that has guided me in the past.
Firstly, It really depends on what you want and what you want to do. First you need to look at your current marketshare versus them and also what you project your future marketshare versus them will be. If you have about equal (or greater) marketshare then getting a minority share is not reasonable and you should negotiate for a fair share. If you have less marketshare than them, and you feel you might be losing traction, then you may want to take a deal with them, working with a lawyer to protect your interests and stake. I'm sure the lawyers will go over this, but you need to ask about anti-dilution clauses, employee termination (fight for termination only via just cause), board seat and/or visitation rights and share with minimal (or no) vesting period.
I would also base the decision on your personal familiarity of developing a business. If you believe you have and can get all the pieces to grow and build the business then I probably wouldn't take the deal, however if you are lacking talent aspects of your business, such as sales and business development, they may be able to provide it for you, or alternatively you can try and recruit a full time biz dev expert.
Also, in my personal experience, at this level litigation is far too costly and you should due your diligence on the company. How do they treat/handle their employees? How do they handle clients and vendors? Do you get a bad feeling that they will try and take advantage of you? Would you get along with them on a day to day? Do you feel that their company will grow yours or that it will be an anchor to you?
One more thing to consider, is the education from the exposure of being in relation with the firm. For instance, I made zero money from the equity from my startup but the education that I received being around business developers, angel networks, VCs and experts in various industries, as well as the contacts was well worth it. It can be like an PhD in entrepreneurship and that can be worth it.
Would love to know more about your specific situation and give some more industry specific advice if you like!
Have a great day and good luck.